PARAMETERIZING FROM THE
EXTREMES: FEASIBLE
PARAMETERIZATIONS OF SOME
NP-OPTIMIZATION PROBLEMS
Somnath Sikdar
The Institute of Mathematical Sciences, Chennai.
A thesis submitted to the
Board of Studies in Mathematical Sciences
in partial fulfilment of the requirements
for the Degree of
Doctor of Philosophy
of
Homi Bhabha National Institute
June 2010
Homi Bhabha National Institute
Recommendations of the Viva Voce Board
As members of the Viva Voce Board, we recommend that the dissertation
prepared by Somnath Sikdar titled Parameterizing From the Extremes: Feasible Parameterizations of Some NP-Optimization Problems be accepted as
fulfilling the requirements for the Degree of Doctor of Philosophy.
Date:
Chair: V. Arvind
Date:
Convener: Venkatesh Raman
Date:
Member 1: Sunil Chandran
Date:
Member 2: Meena Mahajan
Date:
Member 3: K.V. Subrahmanyam
The final approval and acceptance of this dissertation is contingent upon
the candidate’s submission of the final copies of the dissertation to HBNI.
I hereby certify that I have read this dissertation prepared under my direction and recommend that it may be accepted as fulfilling the dissertation
requirement.
Date:
Advisor : Venkatesh Raman
Declaration
I declare that the thesis titled Parameterizing From the Extremes:
Feasible Parameterizations of Some NP-Optimization Problems is a
record of the work carried out by me during the period August 2004
to July 2009 under the supervision of Prof. Venkatesh Raman. This
work is original and it has not been submitted earlier as a whole or
in part for a degree, diploma, associateship or fellowship at this or
any other institute or university.
Chennai,
June 2010.
Somnath Sikdar
Certificate
I certify that the thesis titled Parameterizing From the Extremes:
Feasible Parameterizations of Some NP-Optimization Problems submitted for the degree of Doctor of Philosophy by Somnath Sikdar is
a record of the research carried out by him during the period August
2004 to July 2009 under my supervision. This work has not formed
the basis for the award of any degree, diploma, associateship or fellowship at this or any other institute or university.
Chennai,
June 2010.
Venkatesh Raman
To my family.
I long to accomplish a great and noble task, but
it is my chief duty to accomplish small tasks as
if they were great and noble.
Helen Keller (1880-1968).
Abstract
Parameterized complexity is a newly developed sub-area of computational
complexity that allows for a more refined analysis of problems that are
considered hard in the classical sense. In contrast to the classical theory
where the complexity of a problem is measured in terms of the input size
only, parameterized complexity seeks to exploit the internal structure of a
problem. The complexity of a problem in this case is measured not just in
terms of the input size but in terms of the input size and, what is called,
the parameter.
A parameterized problem is a decision problem whose instances consist
of tuples (I, k), where n = |I| is the size of the input instance and k is the
parameter. The goal here is to design algorithms that decide whether (I, k)
is a yes-instance in time f (k) · nO(1) , where f is a computable function of k
alone, as against a trivial algorithm with running time nk+O(1) . Problems
that admit such algorithms are said to be fixed-parameter tractable and FPT
denotes the class of all fixed-parameter tractable problems. The parameter,
however, is not unique and often there are several ways in which a problem
can be parameterized. This is, in fact, one of the strengths of parameterized
complexity as it allows the same problem to be analyzed in different ways
depending on the parameter. In this thesis, we study different parameterizations of NP-optimization problems with the intent of identifying those
parameterizations that are feasible and most likely to be useful in practice.
A commonly studied parameterization of NP-optimization problems is
the standard parameterized version, where the parameter is the solution
size. We begin by showing that a number of NP-optimization problems,
and in particular problems in MAX SNP, have the property that their optimum solution size is bounded below by an unbounded function of the input
size. We show that the standard parameterized version of these problems is
trivially in FPT and we argue that the natural parameter in such cases is
the deficit between the optimum and the lower bound. That is, one ought
to parameterize above the guaranteed lower bound and we call such a parameterization an “above-guarantee” parameterization. One can similarly
define parameterizations below a guaranteed upper bound. We then introduce the notion of “tight” lower and upper bounds and exhibit problems
for which the above-guarantee and below-guarantee parameterization with
respect to a tight bound is fixed-parameter tractable or W-hard. We show
that if we parameterize “sufficiently” above or below tight bounds, then
these parameterized versions are not in FPT, unless P = NP, for a class of
NP-optimization problems.
We then consider related questions in the approximation algorithms setting. We investigate the possibility of obtaining an approximation algorithm
for an NP-optimization problem that is an ǫ-fraction better than the bestknown approximation ratio for the problem. Since the best-known ratio
could also be the approximation lower-bound for the problem, the algorithm in question could possibly have a worst-case exponential-time complexity. But the challenge is to obtain moderately exponential-time algorithms, whose run-time is possibly a function of ǫ and the input-size, that
deliver (α + ǫ)-approximate solutions. We discuss a technique that allows
us to obtain such algorithms for a class of NP-optimization problems.
We next study the parameterized complexity (and occasionally the approximability) of a number of concrete problems: Kőnig Subgraph problems, Unique Coverage and its weighted variant, a version of the Induced Subgraph problem in directed graphs, and the Directed FullDegree Spanning Tree problem. The Kőnig Subgraph problem is
actually a set of problems where the goal is to decide whether a given graph
has a Kőnig subgraph of a certain size. A graph is Kőnig if the size of a maximum matching equals that of a minimum vertex cover in the graph. Such
graphs have been studied extensively from a structural point-of-view. In this
thesis, we initiate the study of the parameterized complexity and approximability of finding Kőnig subgraphs of a given graph. We will see that one
of the Kőnig Subgraph problems, namely Kőnig Vertex Deletion, is
closely related to a well-known problem in parameterized complexity called
Above Guarantee Vertex Cover. While studying the parameterized
complexity of Kőnig Vertex Deletion, we will also see some interesting
structural relations between matchings and vertex covers of a graph.
Unique Coverage is a natural maximization version of the well-known
Set Cover problem and has applications in wireless networking and radio
broadcasting. It is also a natural generalization of the well-known Max
Cut problem. In this problem we are given a family of subsets of a finite
universe and a nonnegative integer k as parameter, and the goal is to decide
whether there exists a subfamily that covers at least k elements exactly
once. We show that this problem is fixed-parameter tractable by exhibiting
a problem kernel with 4k sets. We also consider a weighted variant of it called
Budgeted Unique Coverage and, by an application of the color-coding
technique, show it to be fixed-parameter tractable.
Our application of color-coding uses an interesting variation of k-perfect
hash families where for every s-element subset S of the universe, and for
every k-element subset X of S, there exists a function that maps X injectively and maps the remaining elements of S into a different range. Such
families are called (k, s)-hash families and were studied before in the context
of coding theory. We prove, using the probabilistic method, the existence
of such hash families of size smaller than that of the best-known s-perfect
hash families. Explicit constructions of such hash families of size promised
by the probabilistic method is open.
We study a version of the Induced Subgraph problem in directed
graphs defined as follows: given a hereditary property P on digraphs, an
input digraph D and a nonnegative integer k, decide whether D has an
induced subdigraph on k vertices with property P. We completely characterize hereditary properties for which this induced subgraph problem is
W[1]-complete for two classes of directed graphs: general directed graphs
and oriented graphs. We also characterize those properties for which the
induced subgraph problem is W[1]-complete for general directed graphs but
fixed-parameter tractable for oriented graphs.
We also study a directed analog of a problem called Full Degree Spanning Tree which has applications in water distribution networks. This
problem is defined as follows: given a digraph D and a nonnegative integer k, decide whether there exists a spanning out-tree of D with at least k
vertices of full out-degree. We show that this problem is W[1]-hard on two
important digraph classes: directed acyclic digraphs and strongly connected
digraphs. In the dual version, called Reduced Degree Spanning Tree,
one has to decide whether there exists a spanning out-tree with at most k
vertices of reduced out-degree. We show that this problem is fixed-parameter
tractable and admits a problem kernel with at most 8k vertices on strongly
connected digraphs and O(k2 ) vertices on general digraphs. We also give an
algorithm for this problem on general digraphs with run-time O∗ (5.942k ).
Acknowledgements
I’ve had the good fortune of meeting a number of wonderful people who
have, over the years, shaped my ideas and helped me in my research career.
I take this opportunity to thank them.
Firstly, I’m indebted to Venkatesh Raman for his guidance, support and
encouragement over the years. He introduced me to the area of parameterized complexity and was extraordinarily patient in explaining his ideas
to me. He was more of a friend to me than a supervisor and this thesis
would certainly not have been possible without his support. I thank Meena
Mahajan for spending so many hours in discussions during my initial years
as a PhD student. She was very patient and helped boost my confidence.
I also thank my doctoral committee members—V. Arvind, Meena Mahajan
and K.V. Subrahmanyam for their helpful advice.
Part of the work for this thesis was supported by DST-DAAD project no.
INT/DAAD/P-138/2006 titled Provably Efficient Algorithms for Computationally Hard Problems between Venkatesh Raman and Rolf Niedermeier.
This included two academic visits to the Friedrich-Schiller University, Jena,
Germany (August 2006–October 2006 and June 2007–July 2007). I thank
Rolf Niedermeier and all the students in the theoretical computer science
group at FSU, Jena, and in particular, Michael Dom and Hannes Moser,
for their hospitality and for making both my stays at Jena such a pleasant
experience. I also thank them for several fruitful discussions we’d had.
I’m grateful to V. Arvind, Kamal Lodaya, Meena Mahajan, Venkatesh
Raman, R. Ramanujam, and C.R. Subramanian for their valuable teaching
which helped me during my formative years. I thank all my teachers at
the Indian Statistical Institute, Kolkata for instilling in me an interest in
theoretical computer science. I thank Mike Fellows for some very inspiring
lectures on parameterized complexity.
I thank all my coauthors for letting me to use results obtained with them
as part of this thesis. I especially thank Saket Saurabh for spending so much
time discussing with me and for some very enjoyable dinners at Jena. I
thank Daniel Lokshtanov, Sounaka Mishra, Neeldhara Misra, Rahul Muthu,
N. Narayanan, Geevarghese Philip, and Srikanth Srinivasan for taking time
out to discuss with me. I thank my batchmates Bireswar Das, Nutan Limaye,
Sunil Simon, and S. Sheerazuddin for all the discussions we’ve had during
the period of our course-work. Some of the people listed here were also part
of the Graph Theory Group which met once a week to discuss problems
from Douglas West’s Introduction to Graph Theory. I thank N.R. Aravind,
Nutan Limaye, Rahul Muthu, N. Narayanan, Saket Saurabh for being such
an enthusiastic part of this group.
During these past years, I was in enjoyable company of friends with whom
I dined frequently. Frequent too were the late-night parties! I thank all the
people who were involved in organizing these parties and for making them
so memorable. I thank Bireswar Das, Pushkar Joglekar, Saptarshi Mandal,
Partha Mukhopadhyay, Rahul Muthu, Prajakta Nimbhorkar, among others
for making my stay at IMSc a pleasant one. I thank the administrative staff
at IMSc for the all the help they provided me during these past years.
Finally, I thank my parents who encouraged me to pursue a career in
research and for their love and support. I thank Satarupa, Mithu, Vinay
and little Aryaman for their love, understanding and encouragement. This
thesis is dedicated to them.
Contents
1 Introduction
1.1 Basic Definitions . . . . . . . . . . . . . . . .
1.1.1 Kernels . . . . . . . . . . . . . . . . .
1.2 Types of Parameterizations . . . . . . . . . .
1.3 Reductions and Parameterized Intractability .
1.4 Notation Used in this Thesis . . . . . . . . .
1.5 Aims and Organization of the Thesis . . . . .
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2 Parameterizing From Default Values
2.1 Why Parameterize From Default Values? . . . . . . . . . .
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Parameterizing From Default Values . . . . . . . . . . . .
2.3.1 Parameterizing Above the 3-Sat Bound . . . . . .
2.3.2 Guarantees Defined by Approximation Algorithms
2.4 Tight Lower and Upper Bounds . . . . . . . . . . . . . . .
2.5 Hard Above or Below-Guarantee Problems . . . . . . . . .
2.6 Parameterizing Sufficiently Beyond Guaranteed Values . .
2.7 Guarantee is a Structural Parameter . . . . . . . . . . . .
2.7.1 Above-Guarantee Vertex Cover . . . . . . . . . . .
2.7.2 The Kemeny Score Problem . . . . . . . . . . . . .
2.8 Conclusion and Further Research . . . . . . . . . . . . . .
2.8.1 Approximating the Above-Guarantee Parameter .
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3 Approximating Beyond the Limit of Approximation
3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . .
3.2 Related Work . . . . . . . . . . . . . . . . . . . . . . .
3.3 Approximating Beyond Approximation Limits . . . . .
3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . .
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4 Kőnig Graphs and Above-Guarantee
4.1 History and Motivation . . . . . . .
4.2 Preliminaries . . . . . . . . . . . . .
4.2.1 Notation . . . . . . . . . . . .
4.2.2 Properties of Kőnig Graphs .
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Vertex Cover
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5 The
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The Above Guarantee Vertex Cover Problem
4.3.1 Parameterized Complexity . . . . . . .
4.3.2 An Approximation Algorithm . . . . .
4.3.3 Hardness of Approximation . . . . . .
The Kőnig Vertex Deletion Problem . . . . .
4.4.1 Parameterized Complexity . . . . . . .
4.4.2 Approximability . . . . . . . . . . . .
The Induced Kőnig Subgraph Problem . . . .
4.5.1 Vertex Induced Kőnig Subgraph . . .
4.5.2 Edge Induced Kőnig Subgraph . . . .
Conclusion and Open Problems . . . . . . . .
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Directed Full Degree Spanning Tree Problem
Problem Definition and Previous Work . . . . . . . . . . .
Digraphs: Basic Terminology . . . . . . . . . . . . . . . .
The d-FDST Problem . . . . . . . . . . . . . . . . . . . .
d-RDST: A Problem Kernel . . . . . . . . . . . . . . . . .
7.4.1 An O(k) Kernel for Strongly Connected Digraphs .
7.4.2 An O(k2 ) Kernel in General Digraphs . . . . . . .
An Algorithm for the d-RDST Problem . . . . . . . . . .
Concluding Remarks . . . . . . . . . . . . . . . . . . . . .
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Unique Coverage Problem
Motivation and Known Results . . . . . . .
Results in this Chapter . . . . . . . . . . .
Unique Coverage: Which Parameterization?
Unique Coverage: The Standard Version . .
5.4.1 Bounded Intersection Size . . . . . .
5.4.2 General Case: A Better Kernel . . .
Budgeted Unique Coverage . . . . . . . . .
5.5.1 Intractable Parameterized Versions .
5.5.2 Parameterizing by both B and k . .
Algorithms for Special Cases . . . . . . . .
5.6.1 Unique Coverage . . . . . . . . . . .
5.6.2 Budgeted Max Cut . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . .
6 The Induced Subgraph Problem
6.1 Problem Definition and Previous Work
6.2 General Directed Graphs . . . . . . . .
6.2.1 W[1]-Completeness Results . .
6.3 Oriented Graphs . . . . . . . . . . . .
6.4 General Digraphs vs Oriented Graphs
6.5 Conclusion . . . . . . . . . . . . . . .
7 The
7.1
7.2
7.3
7.4
7.5
7.6
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Contents
8 Summary and Future Research
8.1 Parameterizing Problems Beyond Default Values
8.2 Kőnig Subgraph Problems . . . . . . . . . . . . .
8.3 Unique Coverage . . . . . . . . . . . . . . . . . .
8.4 NP-Optimization Problems on Directed Graphs .
8.5 Concluding Remarks . . . . . . . . . . . . . . . .
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iv
Contents
List of Figures
4.1
4.2
4.3
An approximation algorithm for AGVC . . . . . . . . . . . .
The sets that appear in the proof of Theorem 4.9 . . . . . . .
The sets that appear in the proof of Lemma 4.10 . . . . . . .
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5.1
The sets in the definition of a (k, s)-hash family. We assume U = [n], C = [t], |S| = s and |X| = k. . . . . . . . . . .
Deterministic algorithm for Budgeted Unique Coverage.
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5.2
7.1
7.2
7.3
The digraph D . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Illustrating the Path Rule: the left and right-hand sides show,
respectively, the situation before and after the transformation. Vertex y1 has two neighbors and vertex y2 just one
neighbor in the set {x1 , . . . , xp−1 }. . . . . . . . . . . . . . . . 124
Algorithm RDST. . . . . . . . . . . . . . . . . . . . . . . . . 134
v
vi
List of Figures
List of Tables
1.1
Complexity of Longest Common Subsequence . . . . . .
6
4.1
Problems dealt with in Chapter 4 . . . . . . . . . . . . . . . .
76
5.1
Main results in this chapter . . . . . . . . . . . . . . . . . . . 103
vii
Chapter 1
Introduction
Parameterized complexity theory is a new branch of complexity theory developed by Downey and Fellows in a series of papers in the early 1990s.
This theory provides a framework for a more refined analysis of algorithmic
problems that are considered hard in the classical sense.
In classical complexity theory, one categorizes problems based on the
amount of a resource, usually time or space, required to solve them. But
how does one measure the amount of a resource? This was answered by
Hartmanis and Stearns who introduced the notion of input size and the idea
of measuring the amount of a resource as a function of the input size [79].
This led to the subsequent development of classical complexity theory as
we know it today with a relatively clean theory of intractability. But if we
measure the complexity of a problem solely in terms of its input size, then
we necessarily ignore all structural information about the input instance in
the resulting theory. This is hardly a desirable situation as the absence
of structural information may make a problem appear harder than what it
actually is. Parameterized complexity theory measures the complexity of a
problem not from the standpoint of the input size alone, but in terms of
both the input size and a parameter. In many situations, the parameter is a
numerical value that captures some structural information about the input
instance. But there are no restrictions on what the parameter can be and
there are cases when the parameter is a structural component of the input,
for example, a graph.
The fundamental notion in parameterized complexity is that of fixedparameter tractability. It extends the classical notion of a feasible algorithm
as one whose running time is polynomial in the input size to admit algorithms whose running time is polynomial in the input size but is possibly
non-polynomial in the parameter. An important contribution of the theory
is the development of a framework where one can show that certain problems
are not fixed-parameter tractable. In this framework, one classifies problems
into complexity classes by means of suitable reductions. Since the theory
is two-dimensional, the reductions and the resulting complexity classes are
1
2
Chapter 1. Introduction
more involved than in the classical case.
In this chapter we introduce the basic definitions in parameterized complexity. We discuss various ways in which parameters can arise and why certain parameterizations are not useful. We also provide a very short survey
of the complexity classes that crop up in parameterized complexity theory.
For a general introduction to this area, one can refer to [47, 59, 107].
1.1
Basic Definitions
We start by defining several key terms used in parameterized complexity.
Let Σ be a nonempty finite alphabet and let Σ∗ denote the set of all finite
strings over Σ. A parameterized problem Q is a subset of Σ × N0 , where N0 is
the set of nonnegative integers. An instance of a parameterized problem is
therefore a pair (I, k), where k is the parameter. Examples of parameterized
problems include:
Example 1.1. Vertex Cover
Input:
An undirected graph G = (V, E) and a nonnegative integer k.
Parameter: The integer k.
Question:
Does there exist a vertex subset S of size at most k such
that every edge of G has an end-point in S?
Example 1.2. Dominating Set
Input:
An undirected graph G = (V, E) and a nonnegative integer k.
Parameter: The integer k.
Question: Does there exist a vertex subset S of size at most k such
that every vertex of G has a neighbor in S?
Example 1.3. Max Sat
Input:
A boolean formula φ in conjunctive normal form with n
variables and m clauses; a nonnegative integer k.
Parameter: The integer k.
Question: Does there exist an assignment that satisfies all but k clauses
of φ?
In the framework of parameterized complexity, the running time of an
algorithm is viewed as a function of two quantities: the size of the problem
instance and the parameter.
1.1. Basic Definitions
3
Definition 1.1. A parameterized problem Q is said to be fixed-parameter
tractable if there exists an algorithm that takes as input an instance (I, k)
of Q and decides whether it is a yes- or a no-instance in time O(f (k)·p(n)),
where n = |I|, f is some computable function and p some polynomial. Such
an algorithm is sometimes called an FPT-algorithm for Q.
The class FPT consists of all fixed-parameter tractable problems. We
assume that an FPT-algorithm for a parameterized problem produces a
witness whenever it outputs yes.
1.1.1
Kernels
Closely related to the notion of an FPT-algorithm is the concept of a kernel.
Definition 1.2. Let Q be a parameterized problem. A kernelization of Q
is a polynomial-time many-one reduction from Q to Q that maps a given
instance (I, k) to an equivalent instance (I ′ , k′ ) such that |I ′ | ≤ g(k) and k′ ≤
h(k), where g and h are two computable functions. The two instances are
equivalent in the sense that (I, k) is a yes-instance if and only if (I ′ , k′ ) is
a yes-instance.
The function g is called the size of the kernel and the polynomial-time
many-one reduction is a kernelization algorithm for Q. A parameterized
problem is kernelizable if it admits a kernelization algorithm.
One reason why the study of kernels is important is because of
Proposition 1.1 ([59]). Let Q be a parameterized problem that is decidable.
Then Q is kernelizable if and only if it is fixed-parameter tractable.
Proof. Suppose Q is kernelizable and let A be a kernelization algorithm
for it. Then the following algorithm is an FPT-algorithm for Q: given
an instance (I, k) of Q, it first computes A(I, k) = (I ′ , k′ ) and then uses
the decision algorithm for Q to decide whether (I ′ , k′ ) ∈ Q. Since |I ′ | ≤
h(k), the running time of the decision algorithm is bounded in terms of the
parameter k.
Conversely suppose that A is an FPT-algorithm for Q with running
time f (k) · |I|c , for some computable f and some c ∈ N. We assume
that ∅ =
6 Q 6= Σ∗ , since otherwise Q has a trivial kernelization that maps
every string in Σ∗ to the empty string ǫ. Fix x0 ∈ Q and x1 ∈ Σ∗ \ Q. The
following algorithm A′ computes a kernel for Q. Given an instance (I, k),
the algorithm A′ simulates the FPT-algorithm A for |I|c+1 steps. If A
stops and accepts (or rejects), then A′ stops and outputs x0 (or x1 , respectively). If A′ does not stop in |I|c+1 steps then it must be that |I| ≤ f (k),
and A′ outputs (I, k). Therefore if (I ′ , k′ ) is the instance output by A′
then |I ′ | ≤ |x0 | + |x1 | + f (k).
4
Chapter 1. Introduction
As the proof of Proposition 1.1 shows, although the notions of kernelizability and fixed-parameter tractability are equivalent, a good FPTalgorithm for a parameterized problem does not necessarily yield a kernel
of small size. In fact, if the parameterized problem is NP-complete, then
any FPT-algorithm for it will run in time superpolynomial (or worse) in the
parameter, assuming P 6= NP. The kernel obtained from such an algorithm
is therefore at least superpolynomial in the parameter and so it is interesting
to ask if the kernel size can be made smaller—in particular, whether it can
be made polynomial in the parameter. It is therefore of independent interest
to consider kernelization algorithms for parameterized problems.
A common technique to establish a problem-kernel is to devise a set of
reduction rules which when applied to the input instance (in some specified
sequence) produces the kernel. Reduction rules may be thought of as steps
in the kernelization algorithm. Formally,
Definition 1.3. A reduction rule for a parameterized problem Q is an algorithm that takes an instance (I, k) of Q and, in time polynomial in |I|
and k, outputs either
1. yes or no, in which case the input instance is a yes- or no-instance,
respectively, or
2. an “equivalent” instance (I ′ , k′ ) of Q such that k′ ≤ k.
Two instances (I, k) and (I ′ , k′ ) are equivalent if the following condition
holds: (I, k) is a yes-instance if and only if (I ′ , k′ ) is a yes-instance.
Definition 1.4. An instance (I, k) of a parameterized problem Q is reduced
with respect to (w.r.t.) a set R of reduction rules if the instance (D ′ , k′ )
output by any reduction rule in R is the original instance (D, k) itself.
As an example of how to use reduction rules to obtain a problem kernel
we consider the Vertex Cover problem. Recall that an instance of this
problem consists of an undirected graph G = (V, E) and a nonnegative
integer k and the question is to decide whether G has a vertex cover of size
at most k. Observe that any vertex of G with degree at least k + 1 must be
in every size k vertex cover of G. Our reduction rules, in this case, take the
following form:
Rule 0. Remove vertices of degree zero from the graph.
Rule 1. If u ∈ V (G) is of degree at least k + 1, set S ← S ∪ {u}, k ← k − 1
and G ← G − u. Return (G, k).
We assume that S is the vertex cover that is being constructed. Here is
a kernelization algorithm for Vertex Cover.
1.2. Types of Parameterizations
5
1. Reduce the input instance (G, k) w.r.t. Rules 0 and 1 to obtain (G′ , k′ ).
If k′ < 0 then G does not have a size k vertex cover and reject.
2. If G′ has more than k′ (k + 1) vertices, reject. Else return (G′ , k′ ).
The bound k′ (k+1) in Step 2 is justified by the fact that a simple undirected
graph with a k′ -vertex cover and degrees bounded above by k and below by 1
has no more than k′ (k + 1) vertices. This algorithm, in polynomial time,
produces an equivalent instance which has at most k2 + k vertices. Buss
and Goldsmith [24] attribute this algorithm to S. Buss. Subsequently Chen
et al. [30] observed that using an analysis of a linear program for Vertex
Cover due to Nemhauser and Trotter [106], one can obtain a kernel with
at most 2k vertices.
Proving polynomial bounds on the size of the kernel for different parameterized problems has been an important practical aspect in the study
of the parameterized complexity of NP-hard problems, and many positive
results are known. See [71] for a survey of kernelization results. Recently
Bodlaender et al. [18] building on the work of Fortnow and Santhanam [63]
developed a lower-bound technique that allows one to prove that a number of parameterized problems do not admit polynomial kernels unless PH
(the polynomial hierarchy) collapses to the third level. Dom et al. [46] have
recently extended the techniques to show many more parameterized problems do not admit polynomial kernels under the same complexity-theoretic
assumptions.
1.2
Types of Parameterizations
We mentioned that the parameter is a numeric value that usually captures
some structural information about the input instance. But there are myriad
ways in which numbers can arise naturally in problem specifications and
therefore the parameter need not be unique. It is often the case that a problem admits several parameterizations and the complexity of these variants
are different from one another. A good example of such a problem is the
Longest Common Subsequence problem which is defined below.
Input:
Question:
A set R of strings r1 , . . . , rl over a finite alphabet Σ and a
nonnegative integer k.
Does there exist a string s ∈ Σ∗ of length at least k such
that s is a subsequence of each string in R?
A string s is a subsequence of r if it can be obtained from r by deleting
some characters from r. Longest Common Subsequence is NP-complete
and the following parameterized variants of it have been studied in the
literature (see [110] for more details).
6
Chapter 1. Introduction
Alphabet size |Σ|
parameter
Parameter
unbounded
l
k
l, k
W[t]-hard (t ≥ 1)
W[2]-hard
W[1]-complete
W[t]-hard (t ≥ 1)
FPT
FPT
constant
W[1]-hard
FPT
FPT
Table 1.1: Complexity of parameterized variants of Longest Common Subsequence.
1. Parameter: l.
2. Parameter: k.
3. Parameter: l, k.
4. Parameter: k, |Σ|.
The parameterized complexity of these variants are summarized in Figure 1.1. The hardness results in the table indicate that the corresponding
parameterized variants are unlikely to be in FPT.
Although one can parameterize an optimization problem in several ways,
it is often the case that the parameter chosen is the solution size and this
particular variant is then called the standard parameterized version. For an
NP-optimization problem Q, the standard parameterized version Qpar is the
following decision problem.
Input:
Question:
A tuple (I, k), where I is an instance of Q and k is a nonnegative integer, the parameter.
Is the optimum solution size of I at least k (if Q is a maximization problem) or at most k (if Q is a minimization
problem)?
Although the standard parameterized version is the most popularly studied
parameterization of NP-optimization problems, we will show in the next
chapter that this version may not always be interesting. In particular, there
are NP-optimization problems for which the standard version is trivially
fixed-parameter tractable.
Consider the (standard parameterization) of Vertex Cover: given a
graph G = (V, E) on n vertices, decide whether G has a vertex cover of
size at most k. The best-known algorithm for Vertex Cover is due to
Chen et al. [30] and runs in time O(1.2852k + kn). Observe that if G has
a matching of size µ then any vertex cover of G picks at least one vertex
from each edge of the matching, and hence is of size at least µ. Therefore
1.3. Reductions and Parameterized Intractability
7
if G has a perfect matching, then for G to have a size k vertex cover we
must have k ≥ n/2. The point being made here is that for a large class
of graphs k = Ω(n), and therefore any algorithm for the standard version
either takes time exponential in n (if k ≥ n/2) or is trivial (if k < n/2).
Now consider an alternative parameterization: does G have a vertex
cover of size at most µ + k? Note that the parameter k here is the size
of the vertex cover above the matching size. Since the matching size is
a guaranteed lower bound on the vertex cover size, we call this version
Above Guarantee Vertex Cover. In this thesis we show that this
version admits an algorithm with running time O(15k · k · m3 ), where m is
the number of edges in the input graph. Note that on the class of graphs
with a perfect matching, this running time is better than any running time
′
of the form O(ck · poly(n)), where k′ = µ + k is the standard parameter.
Just as one can parameterize above guaranteed values, one can consider
parameterizations below guaranteed values. As an example, consider the
following variant of Vertex Cover: given a planar graph G = (V, E)
on n vertices and an integer parameter k, does G admit a vertex cover of
size ⌊3n/4⌋ − k? To see why this is a parameterization below a guaranteed
value, note that any planar graph can be colored using four colors and hence
has an independent set of size at least ⌈n/4⌉, and therefore a vertex cover
of size at most ⌊3n/4⌋. Thus ⌊3n/4⌋ vertices suffice to cover the edges of a
planar graph and this is indeed a parameterization below a guaranteed upper
bound. Also note that a planar graph has a vertex cover of size ⌊3n/4⌋ − k
if and only if it has an independent set of size at least ⌈n/4⌉ + k. The
parameterized complexity of this problem is still open.
In short, parameters can arise in numerous ways and one can conceive
of several different ways to parameterize a given problem. We discuss these
issues in detail in the next chapter and this is in fact one the major themes of
this thesis. For the time being, we turn to a brief discussion on parameterized
intractability.
1.3
Reductions and Parameterized Intractability
Parameterized complexity theory not only provides a rich algorithmic toolkit
to show positive results, but it also provides a framework that allows us to
show that certain problems are unlikely to be fixed-parameter tractable.
As with classical NP-completeness theory, the starting point in the development of any such framework is a suitable notion of a reduction in the
parameterized setting. A reduction in the parameterized setting is defined
as follows:
Definition 1.5. Let L1 and L2 be parameterized languages. We say that L1
fixed-parameter reduces to L2 , denoted by L1 ≤FPT L2 , if there exist functions f, g : N0 → N0 , Φ : Σ∗ × N0 → Σ∗ and a polynomial p(·) such that for
8
Chapter 1. Introduction
any instance (I, k) of L1 ,
1. (Φ(I, k), g(k)) is an instance of L2 computable in time f (k) · p(|I|);
2. (I, k) is a yes-instance of L1 if and only if (Φ(I, k), g(k)) is a yesinstance of L2 .
We say that L1 and L2 are fixed-parameter equivalent if and only if L1 ≤FPT
L2 and L2 ≤FPT L1 .
A fixed-parameter reduction (briefly, an fp-reduction) differs from the
familiar many-one Karp-reduction in two ways. Firstly, the time taken could
be exponential in the parameter k in contrast to a Karp-reduction where the
time taken can be explicitly specified by a polynomial function. Secondly,
the “new” parameter must be a function of the old parameter only, that is,
it should have no dependence on the input size |I|. In a Karp-reduction it
is possible that the new parameter is a function of both k and |I|. It is also
easy to see that if L1 ≤FPT L2 and L2 ∈ FPT then L1 ∈ FPT. That is, the
class FPT is downward-closed under fp-reductions. It is also easy to verify
that an fp-reduction is reflexive and transitive.
If we parallel NP-completeness theory then the other ingredient, besides
the notion of a reduction, is the identification of a class of problems which
is unlikely to be fixed-parameter tractable. In parameterized complexity
theory, this class of problems is defined using a set of satisfiability problems
on Boolean circuits. To describe this class of problems, we first need some
definitions.
A Boolean circuit C with n input nodes is a directed acyclic graph with n
nodes of in-degree zero called input gates, with each input gate labelled by
either a positive literal xi or a negative literal x̄i , where 1 ≤ i ≤ n. The
remaining nodes of the circuit are called gates and are labelled by a Boolean
operator: either And or Or. The circuit has a designated gate of out-degree
zero called the output gate. A circuit C with n input nodes computes an nary Boolean function in a natural way. We say that x ∈ {0, 1}n satisfies C if
the value computed by C on input x is 1. The weight of x is the number of 1s
in x. A gate is called large if its fan-in exceeds some pre-agreed bound > 2;
otherwise it is called small. The weft of a circuit is the maximum number of
large gates on any path from the input gates to the output gate. The depth
of a circuit is a maximum number of gates on any path from the input gates
to the output gate. The size of a circuit C, denoted by |C|, is the total
number of nodes and arcs in it.
We are now ready to define our class of circuit satisfiability problems:
Weighted Weft t Depth h Circuit Satisfiability (WCS(t, h))
Input:
A weft t depth h decision circuit C with n input nodes such
that |C| = poly(n).
Parameter: A nonnegative integer k.
1.3. Reductions and Parameterized Intractability
Question:
9
Does C have a satisfying assignment of weight exactly k?
The pre-agreed bound on the fan-in of small gates is some arbitrary constant. The theory that we describe does not depend on what this bound
actually is. Note that a Boolean 3-Cnf formula (one whose clauses has
at most three literals) can be expressed as a circuit of weft one and depth
two. Therefore Weighted 3-Cnf Satisfiability, which is the problem
of deciding whether a given 3-Cnf formula has a satisfying assignment of
Hamming weight exactly k, is subsumed by WCS(1, 2). Define LC(t,h) to
be the set of tuples (C, k), such that C is a weft t depth h decision circuit
that admits a weight k satisfying assignment. The basic hardness classes of
parameterized complexity theory are defined as follows.
Definition 1.6. A parameterized language L is in the class W[t] if and only
if L fp-reduces to LC(t,h) for some constant h.
We first note that FPT ⊆ W[1]. To see this, let L be a parameterized language in FPT and let (I, k) be an instance of L. One can decide
whether (I, k) is a yes- or no-instance in time f (k) · |I|c , for some computable f and c ∈ N, and accordingly produce a yes- or no-instance, respectively, of Weighted 3-Cnf Satisfiability. This shows that L fp-reduces
to LC(1,2) , and consequently FPT ⊆ W[1]. Since the WCS(t, h) problem
is subsumed by the WCS(t + 1, h) problem, we have W[t] ⊆ W[t + 1] for
all t ≥ 1.
Finally we define:
Weighted Circuit Satisfiability
Input:
A Boolean decision circuit C with n input nodes such
that |C| = poly(n).
Parameter: A nonnegative integer k.
Question: Does C have a satisfying assignment of weight exactly k?
and W[P ] to be the class of parameterized languages that fp-reduce to
Weighted Circuit Satisfiability. Since the set of all polynomial-sized
circuits include all polynomial-sized weft t depth h circuits for all t and h,
we have W[t] ⊆ W[P ] for all t ≥ 1. We therefore obtain an infinite hierarchy
of complexity classes
FPT ⊆ W[1] ⊆ W[2] ⊆ · · · ⊆ W[t] ⊆ · · · ⊆ W[P ]
called the W-hierarchy (W for weft). It is conjectured that these containments are proper.
One can now define the usual notions of hardness and completeness. A
parameterized language L is hard for a class W[t], if all languages in W[t]
10
Chapter 1. Introduction
fp-reduce to L. The language L is W[t]-complete, if L ∈ W[t] and L is hard
for W[t]. Note that, by definition, the language LC(t,h) is complete for the
class W[t] for t ≥ 1.
To understand why one suspects the class W[1] to be the analog of the
class NP, consider the following problem:
Short Turing Machine Acceptance
Input:
A nondeterministic Turing machine M , a string x and an
integer k
Parameter: The integer k.
Question:
Does M have a computation path accepting x in at most k
steps?
This is a parameterized variant of the NP-complete Turing Machine Acceptance problem. To quote Downey and Fellows [47]: “a nondeterministic
Turing machine is such an opaque and generic object that it simply does not
seem reasonable that we should be able to decide in polynomial time whether
a given Turing machine on a given input has some accepting path. It seems
to us that if one accepts the philosophical argument that Turing Machine
Acceptance is intractable, the same reasoning would suggest that Short
Turing Machine Acceptance is fixed-parameter intractable”.
The belief that Short Turing Machine Acceptance is the key fixedparameter intractable problem and the fact that Short Turing Machine
Acceptance is W[1]-complete lend credence to the belief that W[1] is indeed the basic intractability class in the parameterized world. There are
numerous other problems known to be complete for W[1]. We list some of
them in the next theorem. For the definitions of these problems see [47, 19].
Theorem 1.1. [47] The following problems are complete for the class W[1].
1. Short Turing Machine Acceptance.
2. Independent Set.
3. Clique.
4. Weighted c-Cnf Satisfiability, for a fixed c ≥ 1.
There are several problems complete for W[2].
Theorem 1.2. [47] The following problems are complete for the class W[2].
1. Dominating Set.
2. Hitting Set.
3. Set Cover.
1.4. Notation Used in this Thesis
11
Finally, it is interesting to note that there are several natural problems
that are hard for every level of the W-hierarchy.
Theorem 1.3. [19] The following problems are W[t]-hard for all t ≥ 1.
1. Longest Common Subsequence parameterized by the number of
strings.
2. Bandwidth.
3. Perfect Phylogeny.
4. DNA Physical Mapping.
1.4
Notation Used in this Thesis
In this section we fix notation that we use throughout this thesis. Notation
specific to a particular chapter will be described in that chapter itself.
We use two notations to describe the running time of an algorithm. The
first notation we use is the familiar “big-oh” notation. Let f, g : N → N
be functions on natural numbers. We write f (n) = O(g(n)) if there exist
constants c, n0 ∈ N such that for all n ≥ n0 we have f (n) ≤ c · g(n). The
other notation is the O∗ notation and is primarily used for exponentialtime algorithms. If h : N → N is a super-polynomial function on natural
numbers, then O∗ (h(n)) = O(h(n) · p(n)), where p(·) is some polynomial
function. That is, the O ∗ (·) notation suppresses polynomial terms in the
running time expression. From time to time, we will also use the O∗ (·)
notation to describe the running time of a parameterized algorithm. Thus
we write O∗ (f (k)) for a time complexity of the form O(f (k) · nc ), where k
is the parameter, n is the input size and c some constant.
Given an (undirected or directed) graph G, we let V (G) denote its vertex
set and E(G) its edge set. If V ′ ⊆ V (G) then G[V ′ ] denotes the subgraph
of G induced by the vertex set V ′ . We usually reserve the symbols n and m
to denote, respectively, the number of vertices and the number of edges in a
graph. We let N denote the set of natural numbers and N0 the set N ∪ {0}.
The symbol Q denotes the set of rationals and Q≥a , the set of rationals
at least a. For a positive integer n, we let [n] denote the set {1, . . . , n}.
Logarithms to base 2 are denoted by log and to base e by ln.
1.5
Aims and Organization of the Thesis
As discussed in Section 1.2, an NP-optimization problem usually admits
several parameterizations. One of the main aims of this thesis is to study
different parameterizations of NP-optimization problems with the intent of
identifying parameterizations that are most useful from a practical point of
12
Chapter 1. Introduction
view. In Chapters 2 and 3, we try to develop a general theory for parameterizing or approximating a problem beyond a default or best-known bound.
We follow this up (in Chapters 4 through 7) with a study of the parameterized complexity of several concrete NP-optimization problems. We often
consider different ways of parameterizing a problem and investigate how
this changes the (parameterized) complexity of the problem. A detailed
organization of the thesis follows.
In Chapter 2 we study NP-optimization problems which have nontrivial
lower bounds on the size of their optimal solutions. A lower bound is nontrivial if it is an unbounded function of the input size. We show that a number of
NP-optimization problems and, in particular problems in MAX SNP, have
this property and that the standard parameterized version of these problems
is trivially in FPT. We argue that the natural parameter for such problems is
the quantity above the guaranteed lower bound. But parameterizing above
any lower bound may not be interesting, for if a lower bound is “loose” then
even an “above-guarantee” parameterization may trivially be in FPT. This
motivates the introduction of tight lower bounds. Problems parameterized
beyond tight bounds may be in FPT or W-hard, and we give examples of
both. However we show that parameterizations “sufficiently” beyond tight
bounds are not in FPT, unless P = NP, for a large class of NP-optimization
problems.
In Chapter 3, we consider related questions in the approximation algorithms setting. If an NP-optimization problem admits an α-approximation
algorithm, what then is the complexity of obtaining an (α + ǫ)-approximate
solution? Since α could be the approximation lower-bound for the problem,
the algorithm in question could possibly have a worst-case exponential-time
complexity. But the challenge is to obtain moderately exponential-time algorithms, whose running time is possibly a function of ǫ and the input-size,
that deliver (α + ǫ)-approximate solutions. We discuss a technique that allows us to obtain such algorithms for a class of NP-optimization problems.
In Chapter 4, we study the approximation and parameterized complexity of a set of problems where one is required to obtain the largest Kőnig
subgraph of a given graph. A graph G is called Kőnig if the size of a maximum matching is equal to that of a minimum vertex cover of G. One of the
problems we study is Kőnig Vertex Deletion: given a graph G = (V, E)
and a nonnegative integer k, does there exist a set of at most k vertices
whose deletion yields a Kőnig graph? A vertex set which has the property
that deleting it results in a Kőnig graph is called a Kőnig vertex deletion set.
We will see that this problem is closely related to the Above Guarantee
Vertex Cover problem (Section 1.2) and that there are interesting structural relations between vertex covers, matchings and Kőnig vertex deletion
sets.
In Chapter 5 we consider a problem called Unique Coverage which
has applications in wireless networking and radio broadcasting and is also
1.5. Aims and Organization of the Thesis
13
a natural generalization of the well-known Max Cut problem. Like the
Longest Common Subsequence problem (discussed in Section 1.2), this
problem admits several parameterizations and we show that all of them,
except the standard version, are unlikely to be fixed-parameter tractable.
Using techniques from Extremal Combinatorics we show that this problem
has a 4k kernel. We then use the color-coding technique to show that a
weighted variant of this problem, called Budgeted Unique Coverage, is
in FPT. For derandomizing our algorithms, we use k-perfect hash families.
However, we can also use an interesting variation of k-perfect hash families
where, for every s-element subset S of the universe and every k-element
subset X of S, there exists a function that maps X injectively and maps
the remaining elements of S into a different range. We show the existence
of such families of size smaller than the best-known s-perfect hash families
using the probabilistic method. However an explicit construction of such
families of the size promised by the probabilistic method is open.
In Chapters 6 and 7, we study the parameterized complexity of some
NP-optimization problems on directed graphs. In Chapter 6 we consider
the parameterized complexity of the following problem: Given a hereditary
property P on digraphs, an input digraph D and a positive integer k, does D
have an induced subdigraph on k vertices with property P? We completely
characterize hereditary properties for which this induced subgraph problem
is W[1]-complete for two classes of directed graphs: general directed graphs
and oriented graphs. We also characterize those properties for which the
induced subgraph problem is W[1]-complete for general directed graphs but
fixed parameter tractable for oriented graphs.
In Chapter 7 we study a directed analog of the Full Degree Spanning
Tree problem where, given a digraph D and a nonnegative integer k, the
goal is to construct a spanning out-tree of D with at least k vertices of full
out-degree. We show that this problem is W[1]-hard on two basic digraph
classes: directed acyclic digraphs and strongly connected digraphs. In the
dual version, called Reduced Degree Spanning Tree, we parameterize
from the other extreme. Here the goal is to construct a spanning out-tree
with at most k vertices of reduced out-degree. We show that this problem
is fixed-parameter tractable and admits a problem kernel with at most 8k
vertices on strongly connected digraphs and O(k2 ) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with
running time O ∗ (5.942k ).
Finally in Chapter 8 we summarize the results of this thesis and mention
some open problems.
14
Chapter 1. Introduction
Chapter 2
Parameterizing From Default Values
In this chapter we consider parameterizations of NP-optimization problems
with the property that their optimal solution size is either bounded below or
bounded above by an unbounded function of the input size. We argue that
for such optimization problems, the natural parameter is the quantity above
the default lower bound or below the default upper bound. That is, one must
either consider the above-guarantee or below-guarantee parameterizations
relative to the lower or upper bound, respectively.
Apart from showing that there are many natural NP-optimization problems which exhibit this property, we introduce the notion of “tight” upper
and lower bounds and show that parameterizations above or below tight
bounds may be either in FPT or W-hard. Moreover, if we parameterize
“sufficiently” above or below tight bounds, then these parameterized versions are not fixed-parameter tractable unless P = NP, for a subclass of
NP-optimization problems.
2.1
Why Parameterize From Default Values?
Recall from Chapter 1 that the standard parameterized version Qpar of an
NP-optimization problem Q is defined as follows.
Input:
Question:
A tuple (I, k), where I is an instance of Q and k is a nonnegative integer, the parameter.
• Is opt(I) ≥ k? (Q is a maximization problem).
• Is opt(I) ≤ k? (Q is a minimization problem).
Although the standard parameterized version is the most popularly studied
parameterization of NP-optimization problems, there are many problems for
which the standard version is trivially fixed-parameter tractable. Consider
for instance the problem Max c-Sat. An instance of this problem consists of
a Boolean Cnf formula with at most c literals per clause and the objective is
15
16
Chapter 2. Parameterizing From Default Values
to find an assignment which satisfies the maximum number of clauses. This
problem is known to be NP-complete for c ≥ 2. It is well-known that if φ is
a Boolean Cnf formula with m clauses then there exists an assignment that
satisfies at least ⌈m/2⌉ clauses and that such an assignment can be found
in time O(|φ|) (see [104]).
Now consider what this means for the standard parameterized version of
Max c-Sat, for c ≥ 2. An instance of the parameterized version consists of a
tuple (φ, k), where φ is a Boolean Cnf formula with m clauses and at most c
literals per clause and k is a nonnegative integer. The question is whether
there exists an assignment that satisfies at least k clauses of φ. If k ≤ ⌈m/2⌉,
then there exists an assignment which satisfies at least k clauses (because
there is one that satisfies ⌈m/2⌉ clauses) and such an assignment can be
obtained in linear time. We therefore answer yes and output the assignment.
Otherwise, k > ⌈m/2⌉ and since every clause has at most c literals, we
have |φ| ≤ 2kc. We now use a brute-force algorithm that looks at all possible
assignments to the variables of φ (which are at most 2kc in number) and
check whether any of these assignments satisfies at least k clauses of φ.
This brute-force algorithm runs in time O(22kc · |φ|) and is, by definition, an
FPT-algorithm for Max c-Sat. Note, however, that when the brute-force
algorithm is applied, k, and hence the running time is large for all practical
purposes.
A similar state-of-affairs exists for several other problems: Max Cut,
Planar Independent Set and Max Acyclic Subgraph to name a few.
All these problems have some nontrivial lower bound for the optimum value
which is exploited to give a trivial fixed-parameter algorithm for the standard parameterized version of the problem. When the parameter value is
below the default lower bound, the answer is no; otherwise, the standard
brute-force algorithm itself gives a fixed-parameter algorithm since in this
case, the parameter value is considerably large compared to the input size.
However, as we observed before, these algorithms are not necessarily practical.
To deal with this problem, Mahajan and Raman [98] considered a different parameterization of Max Sat and Max Cut, where the parameter is
the difference between the optimum value and the guaranteed lower bound.
They showed that both these problems are FPT under this parameterization as well. More recently, the “above-guarantee” versions of Linear
Arrangement [76] and Minimum Profile [82, 77] have been shown to be
in FPT. Our first observation in this paper is that not just these problems,
but all optimization problems in MAX SNP have a nontrivial lower bound
for the optimum value. We prove that the above-guarantee parameterization
with respect to this lower bound is fixed-parameter tractable. We also observe that approximation algorithms give nontrivial lower or upper bounds
on the solution size. We show that the above or below-guarantee question
with respect to these bounds is fixed-parameter tractable whenever the stan-
2.2. Preliminaries
17
dard parameterized versions of these problems is fixed-parameter tractable.
This is dealt with in Section 2.3 after some definitions in Section 2.2 related
to optimization problems.
We next show that parameterizing above any nontrivial lower bound
may not be interesting because, for some nontrivial lower bounds, the aboveguarantee question may still be trivially FPT. This motivates us to define
what are known as tight lower bounds (Section 2.4).
Tight upper bounds can be analogously defined. Note that most of the
problems we discussed also have nontrivial upper bounds for the optimum
value, and another natural parameterization is to parameterize below the
upper bound. For example, Max Sat has the number of clauses in the
input formula, m, as an upper bound; the upper bound for Max Cut is
the number of edges m in the graph. The natural below-guarantee parameterized questions are: can you satisfy all but k clauses and is there a cut of
size at least m − k? The first is FPT for 2-Cnf Sat [116], and hard (not in
FPT unless P = NP) for c-Cnf Sat for c ≥ 3 [98]. The second problem is
the well-known Odd Cycle Transversal problem which was shown to be
FPT in [117]. Several vertex (edge)-deletion problems, for example, König
Vertex/Edge Deletion [102], Feedback Vertex Set [115], Planar
Vertex Deletion [100], fit in the below-guarantee framework.
In Section 2.5 we show some problems for which the above-guarantee or
below-guarantee version is unlikely to be in FPT. In Section 2.6 we show
that if we parameterize “sufficiently above” tight lower bounds then the
above-guarantee question becomes hard (unless P = NP) for a number of
NP-optimization problems. We show a similar result for the below-guarantee
question with respect to tight upper bounds. Finally in Section 2.8 we list
a number of interesting research directions.
2.2
Preliminaries
An NP-optimization (NPO) problem Q is a four-tuple Q = {I , S, V, opt},
where
1. I is the set of input instances. (Without loss of generality, I can be
recognized in polynomial time.)
2. S(x) is the set of feasible solutions for the input x ∈ I .
3. V is a polynomial-time computable function called the cost function
and for each x ∈ I and y ∈ S(x), V (x, y) ∈ N.
4. opt ∈ {max, min}.
5. The following decision problem Q̃ (called the underlying decision problem of Q) is in NP: Given x ∈ I and an integer k, does there exist
18
Chapter 2. Parameterizing From Default Values
a feasible solution y ∈ S(x) such that V (x, y) ≥ k, when Q is a maximization problem (or, V (x, y) ≤ k, when Q is a minimization problem).
From the above definition, it follows that the optimum value of every
NP-optimization problem is bounded below by 1. We call a lower bound
trivial if it is at most a constant. A nontrivial lower bound for an NPoptimization problem Q is an unbounded function f : N → N such that for
all instances x of Q, we have opt(x) ≥ f (|x|). A nontrivial upper bound
for Q is an unbounded function g : N → N such that for all instances x of Q,
we have opt(x) ≤ g(|x|).
The class MAX SNP was defined by Papadimitriou and Yannakakis [109]
using logical expressiveness. They showed that a number of interesting
optimization problems such as Max 3-Sat, Independent Set-B, Max
Cut, Max k-Colorable Subgraph lie in this class. They also introduced the notion of completeness for MAX SNP by a reduction known as
the L-reduction which we define next.
Let Q1 and Q2 be two optimization (maximization or minimization)
problems. We say that Q1 L-reduces to Q2 if there exist polynomial-time
computable functions f, g, and constants α, β > 0 such that for each instance I1 of Q1 :
1. f (I1 ) = I2 is an instance of Q2 , such that opt(I2 ) ≤ α · opt(I1 ).
2. Given any solution y2 of I2 , g maps (I2 , y2 ) to a solution y1 of I1 such
that |V (I1 , y1 ) − opt(I1 )| ≤ β · |V (I2 , y2 ) − opt(I2 )|
We call such an L-reduction from Q1 to Q2 an (f, g, α, β)-reduction.
An NP-optimization problem Q is MAX SNP-hard if every problem in
the class MAX SNP L-reduces to Q. A problem Q is MAX SNP-complete,
if Q is in MAX SNP and is MAX SNP-hard. Some example MAX SNPcomplete problems are
1. Max c-Sat, for any constant c,
2. Independent Set-B,
3. Vertex Cover-B,
4. Dominating Set-B,
5. Max Cut,
6. Max Directed Cut,
7. Max k-Colorable Subgraph.
2.3. Parameterizing From Default Values
19
Cai and Chen [26] established that the standard parameterized version
of all maximization problems in the class MAX SNP are fixed-parameter
tractable. In the next section, we show that for all problems in MAX SNP,
a certain above-guarantee question is also fixed-parameter tractable.
2.3
Parameterizing From Default Values
The main objective of this section is to show that there exist broad classes
of NP-optimization problems in which every problem has a nontrivial lower
or upper bound on the optimum solution size and that the above or below
guarantee question with respect to these bounds is in FPT. Recall that a
lower or upper bound is nontrivial if it is an unbounded function of the input
size.
We first show that every problem in the class MAX SNP has a nontrivial
lower bound on the optimal solution size and that the parameterized aboveor below-guarantee question with respect to this lower bound is in FPT.
2.3.1
Parameterizing Above the 3-Sat Lower Bound
Consider the problem Max 3-Sat which is complete for the class MAX
SNP. An instance of Max 3-Sat is a Boolean formula f in conjunctive
normal form with at most three literals per clause. As already stated, any
Boolean formula with m clauses has at least ⌈m/2⌉ satisfiable clauses. Using
this, we show the following generalization.
Proposition 2.1. If Q is in MAX SNP, then for each instance I of Q there
exists a positive number γx such that γx ≤ opt(x). Further, if Q is NP-hard,
then the function γ : x → γx is unbounded, assuming P 6= NP.
Proof. Let Q be a problem in MAX SNP and let (f, g, α, β) be an Lreduction from Q to Max 3-Sat. Then for an instance I of Q, f (x) is
an instance of Max 3-Sat such that opt(f (x)) ≤ α · opt(x). If f (x) is a
formula with m clauses, then ⌈m/2⌉ ≤ opt(f (x)) and therefore opt(x) is
bounded below by ⌈m/2⌉/α. This proves that each instance x of Q has a
lower bound. We can express this lower bound in terms of the parameters
of the L-reduction. Since f (x) is an instance of Max 3-Sat, we can take
the size of f (x) to be m. Then γx = |f (x)|/2α = m/2α. Further, note that
if m is not unbounded, then we can solve Q in polynomial time via this
reduction.
Note that this lower bound γx depends on the complete problem to
which we reduce Q. By changing the complete problem, we might construct different lower bounds for the problem at hand. It is also conceivable
that there exist more than one L-reduction between two optimization problems. Different L-reductions could give different lower bounds. Thus the
20
Chapter 2. Parameterizing From Default Values
polynomial-time computable lower bound that we exhibit in Proposition 2.1
is a special lower bound obtained from a specific L-reduction to a specific
complete problem (Max 3-Sat) for the class MAX SNP. Call the lower
bound of Proposition 2.1 a Max 3-Sat-lower bound for the problem Q.
Consider the above-guarantee parameterized version L of Max 3-Sat:
the language L consists of tuples (f, k) such that f is an instance of Max
3-Sat and there exists an assignment satisfying at least k + ⌈m/2⌉ clauses
of the formula f . This problem is fixed-parameter tractable with respect to
parameter k [98] and using this we prove the following.
Theorem 2.1. Let Q be a maximization problem in MAX SNP and let
(f, g, α, β) an L-reduction from Q to Max 3-Sat. For an instance x of Q, let
γx represent the Max 3-Sat-lower bound of x. Then the following problem
is in FPT:
LQ = {(x, k) : x is an instance of Q and opt(x) ≥ γx + k}
Proof. We make use of the fact that there exists a fixed-parameter tractable
algorithm A for Max 3-Sat which takes as input, a pair of the form (ψ, k),
and in time O(|ψ| + h(k)), returns yes if there exists an assignment to the
variables of ψ that satisfies at least ⌈m/2⌉ + k clauses, and no otherwise.
See [98, 108] for such algorithms.
Consider an instance (x, k) of LQ . Then f (x) is an instance of Max 3Sat. Let f (x) have m clauses. Then the guaranteed lower bound for the
instance x of Q, γx = m/2α, and opt(f (x)) ≤ α · opt(x). Apply algorithm
A on input (f (x), kα). If A outputs yes, then opt(f (x)) ≥ m/2 + kα,
implying opt(x) ≥ m/2α + k = γx + k. Thus (x, k) ∈ LQ .
If A answers no, then ⌈m/2⌉ ≤ opt(f (x)) < ⌈m/2⌉ + kα. Apply algorithm A on the inputs (f (x), 1), (f (x), 2), . . . , (f (x), kα), one by one, to obtain opt(f (x)). Let c′ = opt(f (x)). Then use algorithm g of the L-reduction
to obtain a solution to x with cost c. By the definition of L-reduction, we
have |c − opt(x)| ≤ β · |c′ − opt(f (x))|. But since c′ = opt(f (x)), it must
be that c = opt(x). Therefore we simply need to compare c with γx + k to
check whether (x, k) ∈ LQ .
The total time complexity of the above algorithm is O(kα · (|f (x)| +
h(kα)) + p1 (|x|) + p2 (|f (x)|)), where p1 (·) is the time taken by algorithm f
to transform an instance of Q to an instance of Max 3-Sat, and p2 (·) is the
time taken by g to output its answer. Thus the algorithm that we outlined
is indeed an FPT-algorithm for LQ .
Note that the proof of Proposition 2.1 also shows that every minimization
problem in MAX SNP has a Max 3-Sat-lower bound. For minimization
problems whose optimum is bounded below by some function of the input,
it makes sense to ask how far removed the optimum is with respect to the
lower bound. The parameterized question asks whether for a given input x,
opt(x) ≤ γx + k, with k as parameter.
2.3. Parameterizing From Default Values
21
Theorem 2.2. Let Q be a minimization problem in MAX SNP and let
(f, g, α, β) an L-reduction from Q to Max 3-Sat. For an instance x of Q,
let γx represent the Max 3-Sat-lower bound of x. Then the following problem is in FPT:
LQ = {(x, k) : x is an instance of Q and opt(x) ≤ γx + k}.
Proof. As before, let A be an FPT-algorithm for Max 3-Sat which takes
as input, a pair of the form (ψ, k), and in time O(|ψ| + h(k)), returns yes
if there exists an assignment to the variables of ψ that satisfies at least
⌈m/2⌉ + k clauses, and no otherwise. Let (x, k) be an instance of LQ and
f (x) the instance of Max 3-Sat with m clauses. Then γx = m/2α and
opt(f (x)) ≤ α · opt(x). Apply algorithm A on input (f (x), (k + 1) · α).
If A outputs yes, then opt(f (x)) ≥ m/2 + (k + 1) · α, implying opt(x) ≥
m/2α + k + 1 = γx + k + 1. In this case, (x, k) ∈
/ LQ and we return no. If A
answers no, then ⌈m/2⌉ ≤ opt(f (x)) < ⌈m/2⌉ + (k + 1) · α. Apply algorithm
A (k + 1) · α − 1 times on inputs (f (x), 1), (f (x), 2), . . . , (f (x), (k + 1) · α − 1)
to obtain opt(f (x)). Obtain opt(x) as described in the proof of Theorem 2.1
and decide the instance (x, k) appropriately.
Examples of minimization problems in MAX SNP include Vertex Cover-B and Dominating Set-B which are, respectively, the restriction of
the Vertex Cover and the Dominating Set problems to graphs whose
vertex degree is bounded by B.
2.3.2
Guarantees Defined by Approximation Algorithms
If an NP-maximization problem admits an α-approximation algorithm, 0 <
α < 1, then α · opt is a nontrivial polynomial-time computable lower-bound
on the solution size, where opt denotes the optimal solution size. For minimization problems that admit an α-approximation algorithm we have α > 1
and α · opt is a polynomial-time computable upper-bound on the solution size. One can parameterize above or below these nontrivial guaranteed
values. We show that if (1) the NP-optimization problem is polynomially
bounded, (2) its standard parameterized version is in FPT, and (3) α is a
constant, then the α · opt + k question (for maximization problems) and
the α · opt − k question (for minimization problems) are in FPT.
Theorem 2.3. Let Q be an NPO problem which admits a constant-factor
α-approximation algorithm such that its standard parameterized version is
in FPT. Then the following problems are in FPT:
L1 = {(I, k) : max(I) ≥ α · opt + k} if Q is a maximization problem;
L2 = {(I, k) : min(I) ≤ α · opt − k} if Q is a minimization problem.
22
Chapter 2. Parameterizing From Default Values
Proof. Suppose Q is a maximization problem and let (I, k) be an instance
of L1 . Then max(I) ≥ α · opt + k if and only if k ≤ (1 − α) · opt, that is, if
and only if k/(1 − α) ≤ opt. Since the standard parameterized version of Q
is in FPT, deciding whether k/(1 − α) ≤ opt is in FPT. The proof when Q
is a minimization problem is similar.
2.4
Tight Lower and Upper Bounds
For an optimization problem, the question of whether the optimum is at
least “lower bound + k”, for some lower bound and k as parameter, is not
always interesting because if the lower bound is “loose” then the problem
is trivially fixed-parameter tractable. For instance, the following “aboveguarantee” version of Max Cut is trivially in FPT. Given a connected
graph G on m edges, does G have a cut of size at least m/2 + k? By Poljak
and Turzı́k [111], any connected graph G with m edges and n vertices has
a cut of size at least m/2 + ⌈(n − 1)/4⌉, and so if k ≤ ⌈(n − 1)/4⌉, we
answer yes. Otherwise n ≤ 4k and a brute-force algorithm that considers
all possible vertex partitions is an FPT-algorithm.
We therefore examine the notion of a tight lower bound and the corresponding above-guarantee question. A tight lower bound is essentially
the best possible lower bound on the optimum solution size. For the Max
Sat problem, this lower bound is m/2: if φ is an instance of Max Sat,
then opt(φ) ≥ m/2, and there are infinitely many instances for which the
optimum is exactly m/2. This characteristic motivates the next definition.
Definition 2.1 (Tight Lower Bound). A function f : N → N is a tight lower
bound for an NP-optimization problem Q = {I , S, V, opt} if the following
conditions hold:
1. f (|I|) ≤ opt(I) for all I ∈ I .
2. There exists an infinite family of instances I ′ ⊆ I such that opt(I) =
f (|I|) for all I ∈ I ′ .
Note that we define the lower bound to be a function of the input size
rather than the input itself. This is in contrast to the lower bound in Proposition 2.1 which depends on the input instance. We can define the notion of
a tight upper bound analogously.
Definition 2.2 (Tight Upper Bound). A function g : N → N is a tight upper
bound for an NP-optimization problem Q = {I , S, V, opt} if the following
conditions hold:
1. opt(I) ≤ g(|I|) for all I ∈ I .
2. There exists an infinite family of instances I ′ ⊆ I such that opt(I) =
g(|I|) for all I ∈ I ′ .
2.4. Tight Lower and Upper Bounds
23
Some example optimization problems which have tight lower and upper
bounds are given below. The abbreviations tlb and tub stand for tight
lower bound and tight upper bound, respectively.
Max Exact c-Sat
instance A Boolean formula F with n variables and m clauses with
each clause having exactly c distinct literals.
question Find the maximum number of simultaneously satisfiable
clauses.
bounds
tlb = (1 − 1/2c )m;
tub = m.
The expected number of clauses satisfied by the random assignment algorithm is (1 − 1/2c )m; hence the lower bound. To see tightness, note that if
φ(x1 , . . . , xc ) denotes the Exact c-Sat formula comprising of all possible
combinations of c variables, then φ has 2c clauses of which exactly 2c − 1
clauses are satisfiable. By taking disjoint copies of this formula one can
construct Exact c-Sat instances of arbitrary size with exactly (1 − 1/2c )m
satisfiable clauses.
Max Lin-2
instance A system of m linear equations modulo 2 in n variables,
together with positive weights wi , 1 ≤ i ≤ m.
question Find an assignment to the variables that maximizes the total
weight of the satisfied equations.
P
bounds
tlb = W/2, where W = m
tub = W .
i=1 wi ;
If we use {+1, −1}-notation for Boolean values with −1 corresponding
to
Q
true then we can write the ith equation of the system as j∈αi xj = bi ,
where each αi is a subset of [n] and bi ∈ {+1, −1}. To see that we can
satisfy at least half the equations in the weighted sense, we assign values
to the variables sequentially and simplify the system as we go along. When
we are about to give a value to xj , we consider all equations reduced to the
form xj = b, for a constant b. We choose a value for xj satisfying at least
half (in the weighted sense) of these equations. This procedure of assigning
values ensures that we satisfy at least half the equations in the weighted
sense. A tight lower bound instance, in this case, is a system consisting
of pairs xj = bi , xj = b̄i , with each equation of the pair assigned the same
weight. See [81] for more details.
Max Independent Set-B
instance A graph G with n vertices such that the degree of each vertex
is bounded by B.
question Find a maximum independent set of G.
bounds
tlb = n/(B + 1);
tub = n.
24
Chapter 2. Parameterizing From Default Values
A graph whose vertex degree is bounded by B can be colored using B + 1
colors, and in any valid coloring of the graph, the vertices that get the same
color form an independent set. By the pigeonhole principle, there exists
an independent set of size at least n/(B + 1). The complete graph KB+1
on B + 1 vertices has an independence number of n/(B + 1). By taking
disjoint copies of KB+1 one can construct instances of arbitrary size with
independence number exactly n/(B + 1).
Min Dominating Set-B
instance A graph G with n vertices such that the degree of each vertex
is bounded by B.
question Find a minimum dominating set of G.
bounds
tlb = n/(B + 1);
tub = n.
Observe that any vertex can dominate at most B + 1 vertices (including
itself) and thus any dominating set has size at least n/(B + 1). A set of r
disjoint copies of KB+1 has a minimum dominating set of size exactly r =
n/(B + 1). The upper bound of n is met by the class of empty graphs.
Min Vertex Cover-B
instance A graph G with n vertices and m edges such that the degree
of each vertex is at least one and at most B.
question Find a minimum vertex cover of G.
bounds
tlb = m/B;
tub = nB/(B + 1).
Each vertex can cover at most B edges and so any vertex cover has size at
least m/B. This bound is tight for a set of disjoint copies of K1,B ’s. The
upper bound follows from the fact that an independent set in such a graph
has size at least n/(B + 1). The upper bound is met by a disjoint collection
of KB+1 ’s.
Max Planar Independent Set
instance A planar graph G with n vertices and m edges.
question
bounds
Find a maximum independent set of G.
tlb = n/4;
tub = n.
A planar graph is 4-colorable, and in any valid 4-coloring of the graph, the
vertices that get the same color form an independent set. By the pigeonhole
principle, there exists an independent set of size at least n/4. A disjoint set
of K4 ’s can be use to construct arbitrary sized instances with independence
number exactly n/4.
2.4. Tight Lower and Upper Bounds
25
Max Acyclic Digraph
instance A directed graph G with n vertices and m arcs.
question Find a maximum arc-induced acyclic subgraph of G.
bounds
tlb = m/2;
tub = m.
To see that any digraph with m arcs has an acyclic subgraph of size m/2,
place the vertices v1 , . . . , vn of G on a line in that order with arcs (vi , vj ),
i < j, drawn above the line and arcs (vi , vj ), i > j, drawn below the line.
Clearly, by deleting all arcs either above or below the line we obtain an
acyclic digraph. By the pigeonhole principle, one of these two sets must
have size at least m/2. To see that this bound is tight, consider the digraph
D on n vertices: v1 ⇆ v2 ⇆ v3 ⇆ . . . ⇆ vn which has a maximum acyclic
digraph of size exactly m/2. Since n is arbitrary, we have an infinite set of
instances for which the optimum matches the lower bound exactly.
Max Planar Subgraph
instance A connected graph G with n vertices and m edges.
question Find an edge-subset E ′ of maximum size such that G[E ′ ] is
planar.
bounds
tlb = n − 1;
tub = 3n − 6.
Any spanning tree of G has n−1 edges; hence any maximum planar subgraph
of G has at least n − 1 edges. This bound is tight as the family of all trees
achieves this lower bound. An upper bound is 3n − 6 which is tight since
for each n, a maximal planar graph on n vertices has exactly 3n − 6 edges.
Max Cut
instance
question
bounds
A graph G with n vertices, m edges and c components.
Find a maximum cut of G.
tlb = m/2 + ⌈(n − c)/4⌉;
tub = m.
The lower bound for the cut size was proved by Poljak and Turzı́k [111]. This
bound is tight for complete graphs. The upper bound is tight for bipartite
graphs.
Min Linear Arrangement
instance An undirected graph G = (V, E) with n vertices and m
edges.
question Find a one-to-one
P mapping σ : V → {1, 2, . . . , |V |} such that
the function {u,v}∈E |σ(u) − σ(v)| is a minimum.
bounds
tlb = m;
tub = n(n − 1)(n + 1)/6 = n+1
3 .
26
Chapter 2. Parameterizing From Default Values
P
The minimum value of {u,v}∈E |σ(u) − σ(v)| over all one-to-one maps σ
is denoted
P by ola(G). Clearly |σ(u) − σ(v)| ≥ 1 for any mapping σ and
hence {u,v}∈E |σ(u) − σ(v)| ≥ m. This lower bound is tight for the set
of paths. For any n-vertex graph G, ola(G) ≤ ola(Kn ), where Kn denotes
the complete graph on n vertices. It can be easily seen that for the case
of Kn , the value of the objective function remains the same for all one-toone maps σ and that ola(Kn ) = 1 · (n − 1) + 2 · (n − 2) + · · · + (n − 1) · 1 =
n(n − 1)(n + 1)/6.
Min Profile
instance An undirected graph G = (V, E) with n vertices and m
edges.
question Find P
a one-to-one mapping σ : V → {1, 2, . . . , |V |} such
that v∈V prfσ (v) is a minimum, where
prfσ (v) = σ(v) − min{σ(u) : u ∈ N [v]}.
bounds
tlb = m;
tub =
n
2
.
The minimum value of the objective function over all one-to-one maps σ
is denoted by prf(G). This problem is equivalent [17] to the well-known
Interval Graph Completion problem [64, 82] in that if the minimum
number of edges required to be added to a graph to make it interval is k
then its profile is m + k, where m denotes the number of edges in the graph.
Consequently, for any graph G = (V, E),
n
m ≤ prf(G) ≤
.
2
Since for each positive integer m there exists an interval graph with m
edges, this is a tight lower bound on the profile of a graph. Since
complete
n
graphs are interval graphs too, this result also shows that 2 is a tight
upper bound. Interestingly, there is a different lower bound if we restrict
the input graph to be connected. If G is a connected graph on n vertices
then prf(G) ≥ n − 1 [94] and this bound is tight since the profile of a path
on n vertices is n − 1.
Natural questions for maximization problems in the above or belowguarantee framework are whether the languages
La,max = {(I, k) : max(I) ≥ tlb(I) + k}
Lb,max = {(I, k) : max(I) ≥ tub(I) − k}
are in FPT. For minimization problems, one can ask whether the following
are in FPT.
La,min = {(I, k) : min(I) ≤ tlb(I) + k}
Lb,min = {(I, k) : min(I) ≤ tub(I) − k}
2.5. Hard Above or Below-Guarantee Problems
27
The parameterized complexity of such questions are not known for most
problems which are known to have tight bounds. To the best of our knowledge, the above-guarantee question has been shown to be in FPT only for
the Max Sat and Max c-Sat problems [98] and, more recently, for Linear
Arrangement [76] and both versions of Minimum Profile [82, 77].
Since most vertex/edge deletion problems can be cast as below-guarantee
parameterized questions, comparatively more results are known about such
problems. For instance, for the Independent Set problem a trivial upper
bound on the solution size is the number of vertices in the graph. This
bound is tight as the family of trivial graphs meets this bound. Given a
graph G on n vertices, the question of whether there exists k vertices whose
deletion leaves a trivial graph (or equivalently, does G have an independent
set on n − k vertices?) is fixed-parameter tractable, being equivalent to the
well-known Vertex Cover problem [107]. Other examples include Feedback Vertex Set [115], Directed Feedback Vertex Set [31], Odd
Cycle Transversal [117] and Chordal Vertex Deletion [99] which
are all of the form: “is there an acyclic (undirected/directed), bipartite and
chordal graph, respectively, on n − k vertices?”
An interesting example of a non-graph-theoretic problem parameterized
below a guaranteed upper bound is the Min 2-Sat Deletion problem [98].
Min 2-Sat Deletion
Input:
A Boolean 2-CNF formula φ with m clauses and a nonnegative integer k.
Parameter: The integer k.
Question:
Does there exist an assignment that satisfies m − k clauses
of φ?
This problem has been shown to be fixed-parameter tractable by Razgon et
al. [116]. Note that the problem of deciding whether k clauses can be deleted
from a c-Cnf formula to make it satisfiable is NP-hard for c ≥ 3 [98].
In the next section we exhibit problems whose above or below-guarantee
parameterized versions are hard.
2.5
Hard Above or Below-Guarantee Problems
We first exhibit two problems whose above-guarantee parameterized versions
are hard (not in FPT unless P = NP). To the best of our knowledge, these
are the only ones in this category. Consider the problem Min Weight
t-connected Spanning Subgraph defined as follows [34]:
Input:
A connected graph G with n vertices and nonnegative integers t and k.
Parameter: The integer k.
28
Question:
Chapter 2. Parameterizing From Default Values
Does there exists a t-vertex-connected spanning subgraph
of G with at most k edges?
This problem is NP-complete for undirected graphs for t ≥ 2 [34] and is
easily fixed-parameter tractable as shown below.
Lemma 2.1. Let G = (V, E) be a simple, connected graph on n vertices
and let k, t be nonnegative integers. Then deciding whether G has a tvertex-connected spanning subgraph with at most k edges can be done in
time polynomial in n, for every fixed k.
Proof. For t ≥ 2, a t-vertex-connected spanning subgraph of G must have at
least n edges. Therefore if k < n, answer no; else, n ≤ k and any brute-force
algorithm that solves the problem is fixed-parameter tractable.
As discussed in the proof of Lemma 2.1 above, n is a trivial lower bound
for the problem and is tight for the case t = 2. The above-guarantee version,
however, is not fixed-parameter tractable, unless P = NP.
Theorem 2.4. Let G = (V, E) be a simple, connected graph on n vertices
and let k, t be nonnegative integers. Deciding whether G has a t-vertex connected spanning subgraph with at most n + k edges is not fixed-parameter
tractable with respect to parameter k, unless P = NP.
Proof. Note that for t = 2 and k = 0 this problem is equivalent to asking whether G has a Hamiltonian cycle and hence if the above-guarantee
question can be answered in time O(f (k) · nc ), the Hamiltonian Cycle
problem can be solved in polynomial time implying P = NP.
Next consider the Bounded Degree Min Spanning Tree [67] problem.
A connected graph G = (V, E) with edge costs w : E → Z+
and nonnegative integers c and k.
Parameter: The integer k.
Question: Does there exist a spanning subgraph T of total edge-weight
at most k such that each vertex in T has degree at most c?
Input:
Since the total weight of any spanning tree is at least n − 1, we have the
following result.
Lemma 2.2. The Bounded Degree Min Spanning Tree problem is
fixed-parameter tractable with respect to parameter k.
The above-guarantee version (Does G have a spanning tree with total
weight at most n − 1 + k and vertex degrees bounded by c?) is again not
fixed-parameter tractable unless P = NP as the case c = 2 and k = 0 reduces
to solving the Hamiltonian Path problem.
2.6. Parameterizing Sufficiently Beyond Guaranteed Values
29
Theorem 2.5. Given a connected graph G = (V, E) with edge costs w :
E → Z+ and positive integers c and k, deciding whether G has a spanning
tree with weight at most n − 1 + k and vertex degrees bounded by c is not
fixed-parameter tractable with respect to k unless P = NP.
We next consider a problem parameterized below a tight upper bound,
called Wheel-free Deletion, that is W[2]-hard [95]. A vertex v in a
graph G is said to be universal if v is adjacent to all vertices of G. A wheel
is a graph W that has a universal vertex v such that W − v is a cycle. A
graph is wheel-free if no subgraph of G is a wheel. The class of wheel-free
graphs is hereditary [86] and poly-time recognizable [95]. The Wheel-free
Vertex/Edge Deletion problem is defined below:
Input:
A graph G = (V, E) with n vertices and m edges and a
nonnegative integer k.
Parameter: The integer k.
Question:
Does G have an induced subgraph on n − k vertices/m − k
edges that is wheel-free?
Note that for all n and m there exist wheel-free graphs on n vertices or m
edges. The tight upper bound is witnessed by the class of paths, for instance.
In [95], the Wheel-free Vertex/Edge Deletion problems were
shown to be W[2]-hard by a reduction from Hitting Set. This is one
of the few graph-modification problems known to be hard.
2.6
Parameterizing Sufficiently Beyond Default Values
In this section, we study somewhat different, but related, parameterized
questions: Given an NP-maximization problem Q with a tight lower and
a tight upper bound, denoted by tlb and tub, respectively, what is the
parameterized complexity of the following questions?
Qa,max (ǫ) = {(I, k) : max(I) ≥ tlb(I) + ǫ · |I| + k}
Qb,max (ǫ) = {(I, k) : max(I) ≥ tub(I) − ǫ · |I| − k}
(2.1)
(2.2)
Here |I| denotes the input size, ǫ is some fixed positive rational, k is the
parameter and a and b denote, respectively, the above and below-guarantee
version of the problem. For NP-minimization problems, the corresponding
questions are:
Qa,min (ǫ) = {(I, k) : min(I) ≤ tlb(I) + ǫ · |I| + k}
Qb,min (ǫ) = {(I, k) : min(I) ≤ tub(I) − ǫ · |I| − k}
(2.3)
(2.4)
30
Chapter 2. Parameterizing From Default Values
In Theorem 2.6, we show that Problems 2.1 and 2.3 are not fixedparameter tractable for a certain class of problems, unless P = NP. Theorem 2.7 establishes this result for Problems 2.2 and 2.4.
To define the class of optimization problems for which we establish the
hardness result in Theorem 2.6, we need some definitions. To motivate these,
we start with an overview of the proof for maximization problems (Problem 2.1 above). Assume that for some ǫ in the specified range, Qa,max (ǫ) is
indeed in FPT. Now consider an instance (I, s) of the underlying decision
version of Q. Here is a P-time procedure for deciding it. If s ≤ tlb, then
the answer is trivially yes. If s lies between tlb and tlb + ǫ|I|, then “add”
a gadget of suitable size corresponding to the tub, to obtain an equivalent instance (I ′ , s′ ). This increases the input size, but since we are adding
a gadget whose optimum value matches the upper bound, the increase in
the optimum value of I ′ is more than proportional, so that now s′ exceeds
tlb + ǫ|I ′ | and we handle this case next. If s already exceeds tlb + ǫ|I|,
then “add” a gadget of suitable size corresponding to the tlb, to obtain
an equivalent instance (I ′ , s′ ). This increases the input size faster than it
boosts the optimum value of I ′ , so that now s′ exceeds tlb + ǫ|I ′ | by only
a constant, say c1 . Use the hypothesized FPT algorithm for Qa,max (ǫ) with
input (I ′ , c1 ) to correctly decide the original question.
To make this proof idea work, we require that the following conditions
be met:
1. The NPO problem should be such that “addition” of problem instances
is well-defined and that the optimum of the sum is equal to the sum
of the optima (see Definition 2.3).
2. There exist gadgets whose addition to a problem instance increases the
instance size faster than it does the optimum value (see Property P1
below).
3. There exist gadgets whose addition to a problem instance increases the
optimum value faster than it does the instance size (see Property P2
below).
4. The gadgets mentioned in points 2 and 3 must be easily constructible
(see Definition 2.4).
Definition 2.3 (Partially Additive Problems). An NPO problem Q =
{I , S, V, opt} is said to be partially additive if there exists an operator +
which maps a pair of instances I1 and I2 to an instance I1 + I2 such that
1. |I1 + I2 | = |I1 | + |I2 |, and
2. opt(I1 + I2 ) = opt(I1 ) + opt(I2 ).
2.6. Parameterizing Sufficiently Beyond Guaranteed Values
31
A partially additive NPO problem that also satisfies the following condition is said to be additive in the framework of Khanna, Motwani et al. [84]:
there exists a polynomial-time computable function f that maps any solution s of I1 + I2 to a pair of solutions s1 and s2 of I1 and I2 , respectively,
such that V (I1 + I2 , s) = V (I1 , s1 ) + V (I1 , s2 ).
For many graph-theoretic optimization problems, the operator + can be
interpreted as disjoint union. Then the problems Max Cut, Max Independent Set-B, Minimum Vertex Cover, Minimum Dominating Set,
Maximum Directed Acyclic Subgraph, Maximum Directed Cut are
partially additive. For other graph-theoretic problems, one may choose to
interpret + as follows: given graphs G and H, G + H refers to a graph obtained by placing an edge between some (possibly arbitrarily chosen) vertex
of G and some (possibly arbitrarily chosen) vertex of H. The Max Planar
Subgraph problem is partially additive with respect to both these interpretations of +. For Boolean formulae φ and ψ in conjunctive normal form
with disjoint sets of variables, define + as the conjunction φ ∧ ψ. Then the
Max Sat problem is easily seen to be partially additive.
Definition 2.4 (Dense Set). Let Q = {I , S, V, opt} be an NPO problem. A
set of instances I ′ ⊆ I is said to be dense with respect to a set of conditions
C if there exists a constant c ∈ N such that for all closed intervals [a, b] ⊆ R+
of length |b − a| ≥ c, there exists an instance I ∈ I ′ with |I| ∈ [a, b] such
that I satisfies all the conditions in C. Further, if such an I can be found
in polynomial time (polynomial in b), then I ′ is said to be dense poly-time
uniform with respect to C.
For example, for the Maximum Acyclic Digraph problem, the set of
all oriented digraphs (digraphs without 2-cycles) is dense (poly-time uniform) with respect to the condition: opt(G) = |E(G)|.
Let Q = {I , S, V, opt} be an NP-optimization problem with a tight
lower bound f : N → N and a tight upper bound g : N → N. We assume
that both f and g are increasing and satisfy the following conditions
P1 For all a, b ∈ N, f (a + b) ≤ f (a) + f (b) + c∗ , where c∗ is a constant
(positive or negative),
P2 There exists n0 ∈ N and r ∈ Q+ such that g(n) − f (n) > rn for all
n ≥ n0 .
Property P 1 is satisfied by linear functions (f (n) = an + b) and by some
√
sub-linear functions such as n, log n, 1/n. Note that a super-linear function
cannot satisfy P 1.
Now that we have formally defined all the required properties, we can
state the theorem precisely.
Theorem 2.6. Let Q = {I , S, V, opt} be a polynomially bounded NPO
problem such that the following conditions hold.
32
Chapter 2. Parameterizing From Default Values
1. Q is partially additive.
2. Q has a tight lower bound (tlb) f , which is increasing and satisfies
condition P 1. The infinite family of instances I ′ witnessing the tight
lower bound is dense poly-time uniform with respect to the condition
opt(I) = f (|I|).
3. Q has a tight upper bound (tub) g, which with f satisfies condition P 2.
The infinite family of instances I ′ witnessing the tight upper bound is
dense poly-time uniform with respect to the condition opt(I) = g(|I|).
4. The underlying decision problem Q̃ of Q is NP-hard.
Let p := sup {r ∈ Q+ : g(n) − f (n) > rn for all n ≥ n0 } and for 0 < ǫ < p,
define Qa (ǫ) to be the following parameterized problem
Qa (ǫ) = {(I, k) : opt(I) ≥ f (|I|) + ǫ|I| + k} for max problems;
Qa (ǫ) = {(I, k) : opt(I) ≤ f (|I|) + ǫ|I| + k} for min problems.
If Qa (ǫ) is FPT for any 0 < ǫ < p, then P = NP.
Proof. We present a proof for NP-maximization problems and towards the
end we outline the necessary changes needed for this proof to work for minimization problems. Therefore let Q be an NP-maximization problem and
suppose that for some 0 < ǫ < p, the parameterized problem Qa (ǫ) is fixedparameter tractable. Let A be an FPT-algorithm for it with run time
O(t(k)poly(|I|)). We will use A to solve the underlying decision problem of
Q in polynomial time. Note that for 0 < ǫ < p, g(n) − f (n) − ǫn is strictly
increasing and strictly positive for large enough values of n.
Let (I, s) be an instance of the decision version of Q. Then (I, s) is a
yes-instance if and only if max(I) ≥ s. We consider three cases and proceed
as described below.
Case 1: s < f (|I|).
Since max(I) ≥ f (|I|), we answer yes.
Case 2: f (|I|) ≤ s < f (|I|) + ǫ|I|.
In this case, we claim that we can transform the input instance (I, s)
into an ‘equivalent’ instance (I ′ , s′ ) such that
1. f (|I ′ |) + ǫ|I ′ | ≤ s′ .
2. |I ′ | = poly(|I|).
3. opt(I) ≥ s if and only if opt(I ′ ) ≥ s′ .
This will show that we can, without loss of generality, go to Case 3 below
directly.
2.6. Parameterizing Sufficiently Beyond Guaranteed Values
33
To achieve the transformation, add a tub instance I1 to I. Define I ′ =
I + I1 and s′ = s + g(|I1 |). Then it is easy to see that max(I) ≥ s if and
only if max(I ′ ) ≥ s′ . We want to choose I1 such that f (|I ′ |) + ǫ|I ′ | ≤ s′ .
Since |I ′ | = |I| + |I1 | and s′ = s + g(I1 ), and since f (|I|) < s, it suffices to
choose I1 satisfying
f (|I| + |I1 |) + ǫ|I| + ǫ|I1 | ≤ f (|I|) + g(|I1 |)
By Property P 1, we have f (|I| + |I1 |) ≤ f (|I|) + f (|I1 |) + c∗ , so it suffices
to satisfy
f (|I1 |) + c∗ + ǫ|I| + ǫ|I1 | ≤ g(|I1 |)
By Property P2 we have g(|I1 |) > f (|I1 |) + p|I1 |, so it suffices to satisfy
c∗ + ǫ|I| ≤ (p − ǫ)|I1 |
Such an instance I1 (of size polynomial in |I|) can be chosen because 0 <
ǫ < p, and because the tight upper bound is polynomial-time uniform dense.
Case 3: f (|I|) + ǫ|I| ≤ s
In this case, we transform the instance (I, s) into an instance (I ′ , s′ ) such
that
1. f (|I ′ |) + ǫ|I ′ | + c1 = s′ , where 0 ≤ c1 ≤ c0 and c0 is a fixed constant.
2. |I ′ | = poly(|I|).
3. max(I ′ ) ≥ s′ if and only if max(I) ≥ s.
We then run algorithm A with input (I ′ , c1 ). Algorithm A answers yes if
and only if max(I ′ ) ≥ s′ . By condition 3 above, this happens if and only if
max(I) ≥ s. This takes time O(t(c1 ) · poly(|I ′ |)).
We obtain I ′ by adding a tlb instance I1 to I. What if addition of any
tlb instance yields an I ′ with s′ < f (I ′ ) + ǫ|I ′ |? In this case, s must already
be very close to f (|I|) + ǫ|I|; the difference k , s − f (|I|) − ǫ|I| must be
at most ǫd + c∗ , where d is the size of the smallest tlb instance I0 . (Why?
Add I0 to I to get s + f (d) < f (|I| + d) + ǫ(|I| + d); applying property P1,
we get s + f (d) < f (|I|) + f (d) + c∗ + ǫ|I| + ǫd, and so k < c∗ + ǫd.) In
such a case, we can use the FPT algorithm A with input (I, k) directly to
answer the question “Is max(I) ≥ s?” in time O(t(ǫd + c∗ ) · poly(|I|)).
So now assume that k ≥ c∗ + ǫd, and it is possible to add tlb instances
to |I|. Since f is an increasing function, there is a largest tlb instance I1 we
can add to I to get I ′ while still satisfying s′ ≥ f (I ′ ) + ǫ|I ′ |. The smallest
tlb instance bigger than I1 has size at most |I1 | + c, where c is the constant
that appears in the definition of density. We therefore have the following
inequalities
f (|I ′ |) + ǫ|I ′ | ≤ s′ < f (|I ′ | + c) + ǫ(|I ′ | + c).
34
Chapter 2. Parameterizing From Default Values
Since f is increasing and satisfies property P1, we have
f (|I ′ | + c) + ǫ(|I ′ | + c) − f (|I ′ |) + ǫ|I ′ | ≤ f (c) + c∗ + ǫc , c0 ,
and hence s′ = f (|I ′ |) + ǫ|I ′ | + c1 , where 0 ≤ c1 ≤ c0 . Note that c0 is
a constant independent of the input instance (I, s). Also, since Q is a
polynomially bounded problem, |I1 | is polynomially bounded in |I|.
Note that the proof for Cases 2 and 3 do not make explicit use of the fact
that Q is a maximization problem; the proof here goes through for minimization problems as well. In fact, the only change necessary for minimization
problems is in Case 1 where if s < tlb, we return no.
Remark 2.1. Note that there are some problems, notably Max 3-Sat, for
which the constant c0 in Case 3 of the proof above, is 0. For such problems,
the proof of Theorem 2.6 actually proves that the problem Q′ = {(I, k) :
max(I) ≥ f (|I|) + ǫ|I|} is NP-hard. But in general, the constant c0 ≥ 1 and
so this observation cannot be generalized.
The constraints imposed in Theorem 2.6 seem to be rather strict, but
they are satisfied by a large number of NP-optimization problems.
Corollary 2.1. For any NP-optimization problem Q in the following list,
the Qa (ǫ) problem is not fixed-parameter tractable unless P = NP:
Problem
tlb(I) + ǫ · |I| + k
Range of ǫ
Max Sat
Max c-Sat
Max Exact c-Sat
Max Lin-2
Planar Independent Set
( 21 + ǫ)m + k
( 21 + ǫ)m + k
(1 − 21c + ǫ)m + k
( 12 + ǫ)m + k
( 41 + ǫ)n + k
0<ǫ<
0<ǫ<
0<ǫ<
0<ǫ<
0<ǫ<
6. Independent Set-B
1
( B+1
+ ǫ)n + k
0<ǫ<
7. Dominating Set-B
1
+ ǫ)n +
( B+1
m
B + ǫn + k
( 21 + ǫ)m + k
0<ǫ<
1.
2.
3.
4.
5.
8.
9.
10.
11.
Vertex Cover-B
Max Acyclic Subgraph
Max Planar Subgraph
Max Cut
12. Max Dicut
k
(1 + ǫ)n − 1 + k
m
n−c
2 + ⌈ 4 ⌉ + ǫn + k
q
1
1
m
m
4 +
32 + 256 − 16 + ǫm + k
1
2
1
2
1
2c
1
2
3
4
B
B+1
B
B+1
0 < ǫ < h(B)
0 < ǫ < 12
0<ǫ<2
0 < ǫ < 14
0<ǫ<
3
4
Remark 2.2. Note that for Vertex Cover-B, the lower bound is expressed in terms of the number of edges m whereas the question asked is
whether there exists a cover of size at most tlb + ǫ · n + k. For Max Cut,
the lower bound is in terms of the number of vertices n and edges m but the
question asked is whether there exists a cut of size at least tlb + ǫ · n + k. In
both these cases, the parameter used to express the input size in the “ǫ·f (|I|)
2.6. Parameterizing Sufficiently Beyond Guaranteed Values
35
part” is not the same as that in the expression for the lower bound. Hence
these do not satisfy the conditions of Theorem 2.6 but can be proved independently using the same proof-technique. The function h(B) in Item 8 of
Corollary 2.1 is given by
h(B) =
(B − 1)(2B + 1)
.
2B(B + 1)
We now extend Theorem 2.6 to the corresponding variant of the belowguarantee question. Let Q = {I , S, V, opt} be an NPO problem with a
tight lower bound f : N → N and a tight upper bound g : N → N which are
increasing functions and satisfy the following conditions
P3 For all a, b ∈ N, g(a + b) ≤ g(a) + g(b) + c∗ , where c∗ is a constant,
P4 There exists r ∈ Q+ such that g(n) − f (n) > rn for all n ≥ n0 for
some n0 ∈ N.
Theorem 2.7. Let Q = {I , S, V, opt} be a polynomially bounded NPO
problem such that the following conditions hold.
1. Q is partially additive.
2. Q has a tight lower bound (tlb) f such that the infinite family of instances I ′ witnessing the tight lower bound is dense poly-time uniform
with respect to the condition opt(I) = f (|I|).
3. Q has a tight upper bound (tub) g which is increasing, satisfies condition P 3, and with f satisfies P 4. The infinite family of instances I ′
witnessing the tight upper bound is dense poly-time uniform with respect to the condition opt(I) = g(|I|).
4. The underlying decision problem Q̃ of Q is NP-hard.
Let p := sup {r ∈ Q+ : g(n) − f (n) > rn for all n ≥ n0 } and for 0 < ǫ < p,
define Qb (ǫ) to be the following parameterized problem
Qb (ǫ) = {(I, k) : max(I) ≥ tub(I) − ǫ · |I| − k} (max problems);
Qb (ǫ) = {(I, k) : min(I) ≤ tub(I) − ǫ · |I| − k} (min problems).
If Qb (ǫ) is FPT for any 0 < ǫ < p, then P = NP.
Proof Sketch. We sketch a proof for NP-minimization problems. Assume
that for some ǫ in the specified range, Qb (ǫ) is indeed in FPT. Consider
an instance (I, s) of the underlying decision version of Q. Here is a P-time
procedure for deciding it. If s > tlb, then the answer is trivially yes. If
s lies between tub and tub − ǫ|I|, then “add” a gadget of suitable size
corresponding to the tlb to obtain an equivalent instance (I ′ , s′ ). This
36
Chapter 2. Parameterizing From Default Values
increases the input size, but since we are adding a gadget whose optimum
value matches the lower bound, the increase in the optimum value of I ′ is less
than proportional, so that now s′ is less than tub−ǫ|I ′ |. If s were already less
than tub−ǫ|I|, then “add” a gadget of suitable size corresponding to the tub
to obtain an equivalent instance (I ′ , s′ ). This increases the optimum value
faster than it does the instance size, so that now s′ is less than tub − ǫ|I ′ |
by only a constant, say c1 . Use the hypothesized FPT algorithm for Qb (ǫ)
with input (I ′ , c1 ) to correctly decide the original question.
2
The next result shows that for a number of NP-optimization problems,
the below-guarantee parameterized variant is unlikely to be in FPT.
Corollary 2.2. For any NP-optimization problem Q in the following list,
the Qb (ǫ) problem is not fixed-parameter tractable unless P = NP:
1.
2.
3.
4.
5.
Problem
tub(I) − ǫ · |I| − k
Range of ǫ
Max Sat
Max c-Sat
Max Exact c-Sat
Max Lin-2
Planar Independent Set
(1 − ǫ)m − k
(1 − ǫ)m − k
(1 − ǫ)m − k
(1 − ǫ)m − k
(1 − ǫ)n − k
0<ǫ<
0<ǫ<
0<ǫ<
0<ǫ<
0<ǫ<
(1 − ǫ)n − k
0<ǫ<
(1 − ǫ)n − k
0<ǫ<
8. Vertex Cover-B
B
( B+1
0<ǫ<
9.
10.
11.
12.
(1 − ǫ)m − k
(3 − ǫ)n − 6 − k
m − ǫn − k
(1 − ǫ)m − k
6. Independent Set-B
7. Dominating Set-B
Max
Max
Max
Max
Acyclic Subgraph
Planar Subgraph
Cut
Dicut
− ǫ)n − k
1
2
1
2
1
2c
1
2
3
4
B
B+1
B
B+1
(B−1)(2B+1)
2B(B+1)
1
2
0<ǫ<
0<ǫ<2
0 < ǫ < 41
0 < ǫ < 34
Remark 2.3. For Max Cut, the upper bound is expressed in terms of the
number of edges, whereas the parameter used to express the input size in
the “ǫ · g(|I|) part” is the number of vertices. This again does not satisfy
the conditions of Theorem 2.7 but can be proved independently using the
same proof-technique.
2.7
When the Guarantee is a Structural Parameter
Thus far, we examined the situation when the lower bound is a function
of the input size. We now consider problems where the lower bound is a
function of the problem instance rather than the input size.
2.7.1
Above-Guarantee Vertex Cover
In Chapter 1 we discussed the standard and above-guarantee parameterizations of Vertex Cover. To recapitulate, the standard parameterized
2.7. Guarantee is a Structural Parameter
37
version (given a graph G = (V, E), does it have a vertex cover of size at
most k?) is fixed-parameter tractable by a number of algorithms (see [107]).
However for a given instance (G, k) of the problem to be a yes-instance,
the standard parameter k must be at least the size of a maximum matching
in G. Thus k = Ω(|V |) in general and any FPT-algorithm for the standard
version takes time exponential in |V |. It is therefore natural to consider the
above-guarantee version of Vertex Cover.
Above-Guarantee Vertex Cover
Input:
A graph G with a maximum matching of size µ.
Parameter: A nonnegative integer k.
Question:
Does G have a vertex cover of size at most µ + k?
Note that the lower bound guarantee for this problem is different in the
sense that it is a function of the input instance and not the input size. The
Above-Guarantee Vertex Cover is fixed-parameter reducible [102] (in
fact, fixed-parameter equivalent [57]) to Min 2-Sat Deletion. See Chapter 4 for more details.
The parameterized complexity of these problems was open for quite some
time until recently Razgon et al. [116] showed the latter problem to be fixedparameter tractable. This also shows that Above-Guarantee Vertex
Cover is fixed-parameter tractable. Interestingly, it was already known
that one can check in polynomial time whether the size of a minimum vertex cover equals that of a maximum matching [44]. This is in contrast to
the problems considered before (problems in Section 2.4) where we do not
know whether there exists a polynomial time algorithm to decide whether a
given instance has an optimum value equal to the tight lower bound. For a
discussion on this issue see Section 2.8.
2.7.2
The Kemeny Score Problem
The Kemeny Score problem is a rank-aggregation problem that arises
in social choice theory [49, 15]. Informally, the goal of this problem is to
combine a number of different rank orderings on the same set of candidates
to obtain a “best” ordering. An election (V, C) consists of a set V of votes
and set C of candidates. A vote is simply a preference list of candidates (a
permutation of C). A “Kemeny consensus” is a preference list of candidates
that is “closest” to the given set of votes. Given a pair of votes π1 , π2 , the
Kendall-Tau-distance (KT-distance for short) between π1 and π2 is defined
as
X
dist(π1 , π2 ) =
dπ1 ,π2 (c, d),
{c,d}⊆C
where dπ1 ,π2 (c, d) = 0 if π1 and π2 rank c and d in the same order, and 1
otherwise. The score of a preference list π with respect to an election (V, C)
38
Chapter 2. Parameterizing From Default Values
is defined as
scr(π) =
X
dist(π, πi ).
πi ∈V
A preference list π with minimum score is called a Kemeny consensus of the
election (V, C) and its score scr(π) is called the Kemeny score of (V, C).
The Kemeny Score problem is the following.
Kemeny Score
Input:
An election (V, C) and an integer k.
Parameter: The integer k.
Question:
Is the Kemeny score of (V, C) at most k?
This problem is NP-complete even for the case when the number of votes
is four, whereas the complexity of the case |V| = 3 is still open [49]. The
case |V| = 2 can be solved trivially as the Kemeny score is simply the KTdistance of the two votes. In [15], Kemeny Score was shown to be in
FPT. Also note that there are at most |C|! distinct votes and therefore for
constant |C|, the problem is polynomial-time solvable irrespective of the
number of votes.
Observe that given an election (V, C) and a preference list π, the score
of π with respect to (V, C) is lower bounded by
X
LV,C =
min{ν(a, b), ν(b, a)},
{a,b}∈C
where ν(a, b) is the number of preference lists in V that rank a higher than b.
Thus the Kemeny score of (V, C) is lower bounded by LV,C . Observe that
this lower bound is tight since in the case where V contains |V| − 1 identical
preference lists along with a single copy of the reverse of these lists, the
Kemeny score of (V, C) equals LV,C = |C|
2 .
Hence a more natural question is:
Above-Guarantee Kemeny Score
Input:
An election (V, C) and an integer k.
Parameter: The integer k.
Question: Is the Kemeny score of (V, C) at most LV,C + k?
We note that this problem is fixed-parameter tractable by a parameterpreserving reduction to a weighted variant of Directed Feedback Vertex Set, where the vertices have weights and one has to decide whether
there exists a feedback vertex set of weight at most k, with k as parameter.
The following reduction appears in [49]. Given an election (V, C), construct an arc-weighted directed graph G on |C| vertices as follows: for each
2.8. Conclusion and Further Research
39
pair of vertices u and v, there exists an arc from u to v if and only if the
majority of the votes in V rank u higher than v; the weight of the arc from u
to v, w(u, v), is the difference between the number of votes that rank u
higher than v and vice versa. Note that in case of tie, there are no arcs
between u and v.
Lemma 2.3. The election (V, C) has a Kemeny score of at most LV,C + k
if and only if G has a feedback arc set of weight at most k.
Proof. Assume that the election (V, C) has a Kemeny score of at most LV,C +
k and let π be a permutation that attains this score. Let S be the set of
arcs (b, a) ∈ A(G) such that P
π ranks a before b (but the majority of lists in V
rank b before a). Then k ≤ (b,a)∈S w(b, a). We claim that if we delete the
arcs in S, then the resulting digraph is acyclic. For if (a1 , a2 , . . . , ak , ak+1 =
a1 ) is a dicycle in G, then at least one of the arcs (ai , ai+1 ) will have been
deleted. For otherwise, in the permutation π, we would have ai preceding ai+1 for all 1 ≤ i ≤ k, an impossibility.
Conversely, suppose that G has a feedback arc set T of weight at most k.
Let π be a topological ordering of the vertices in the DAG obtained by
deleting the arc-set T from G. We claim that scr(π) ≤ LV,C + k. If π
ranks a before b then the contribution of the pair {a, b} to scr(π) is ν(b, a).
If (a, b) ∈ E(G) then ν(b, a) = min{ν(a, b), ν(b, a)}; if (b, a) ∈ E(G) then
ν(b, a) = w(b, a) + ν(a, b) = w(b, a) + min{ν(a, b), ν(b, a)}.
Note that the arcs in T are precisely those
P arcs (b, a) ∈ E(G) such that π
ranks a before b and therefore k =
π(a)<π(b) w(b, a). Hence scr(π) =
P
ν(b,
a)
=
L
+
k.
V,C
π(a)<π(b)
The problem of deciding whether an arc-weighted directed graph has a
feedback arc set of size at most k fixed-parameter reduces to the problem
of deciding whether a vertex-weighted directed graph has a feedback vertex
set of size at most k [53]. This latter problem is in FPT since the algorithm
for Directed Feedback Vertex Set presented in [31] actually works for
this vertex-weighted variant of the problem. Consequently,
Theorem 2.8. Above-Guarantee Kemeny Score is in FPT.
Open Problem 2.1. Are there other “natural” problems where the lower
bound is a function of the input instance rather than the input size? If so,
are their above guaranteed versions fixed-parameter tractable?
2.8
Conclusion and Further Research
We have argued that for several optimization problems including all those
in MAX SNP, the above or below-guarantee parameterization is the natural
40
Chapter 2. Parameterizing From Default Values
and more practical direction to pursue. In Section 2.5 we exhibited two
problems for which the above-guarantee parameterization is hard and there
are problems such as Max Sat [98], Min Linear Arrangement [76] and
Min Profile [82, 77] for which this question is fixed-parameter tractable.
The main problem left open is:
Open Problem 2.2. Is there a characterization for the class of problems
for which the above or below-guarantee question with respect to a tight
lower or upper bound is in FPT (or W[1]-hard)?
We believe that there are several natural directions to pursue both from
an algorithmic as well as from a practical point of view. As stated before,
not many results are known on parameterized above or below-guarantee
problems. In fact, the complexity of problems (1) through (7) stated in
Section 2.4, when parameterized above their guaranteed values, is open.
Some of the more interesting above-guarantee problems are:
Open Problem 2.3. Planar Independent Set: Given an n-vertex planar graph and an integer parameter k, does G have an independent set of
size at least ⌈n/4⌉ + k?
Open Problem 2.4. Max Cut: Given a connected graph G on n vertices
and m edges and a parameter k, does G have a cut of size at least ⌈m/2⌉ +
⌈(n − 1)/2⌉/2 + k?
Open Problem 2.5. Max Exact c-Sat: Given a Boolean CNF formula F
with m clauses such that each clause has exactly c distinct literals and
an integer parameter k, does there exist an assignment that satisfies at
least (1 − 2−c )m + k clauses?
In all the above problems, the case k = 0 is interesting in itself since an
FPT-algorithm for any of the above problems implies a polynomial-time test
for deciding whether the optimum equals the lower bound for that particular
problem. In fact, one way to prove that an above-guarantee problem does
not have an FPT-algorithm is by showing that there is no polynomial time
algorithm (assuming P 6= NP) that decides whether the optimum equals the
guaranteed lower bound.
Open Problem 2.6. Is there a polynomial time algorithm that decides
whether
1. a given planar graph G on n vertices has a maximum independent set
of size exactly ⌈n/4⌉?
2. a given connected graph G with m edges and n vertices has a maximum
cut of size exactly ⌈m/2⌉ + ⌈(n − 1)/2⌉/2?
2.8. Conclusion and Further Research
41
3. a given Boolean CNF formula F with m clauses with exactly c distinct literals per clause, has a maximum of (1 − 2−c )m simultaneously
satisfiable clauses?
Here are some interesting below-guarantee problems which are open:
Open Problem 2.7. [102] König Edge Deletion Set: Given a graph G
on n vertices, m edges and an integer k, does there exist an edge-induced
subgraph of G on m − k edges that is König (a graph in which a minimum
vertex cover and a maximum matching have the same size)? In other words,
can k edges be deleted from G to make it König? In Chapter 4 we show
that the related vertex version is in FPT.
Open Problem 2.8. [36, 105] Perfect Vertex Deletion: Let G be a
graph on n vertices and m edges and k a nonnegative integer. Does there
exist a vertex-induced subgraph on n − k vertices that is perfect? A similar
question can be framed for the edge version.
2.8.1
Approximating the Above-Guarantee Parameter
We examine the notion of an above-guarantee approximation algorithm. The
idea, as in recent attempts to study parameterized approximation [27, 48,
33], is to try and approximate the parameter k by an efficient algorithm.
To motivate this discussion, we consider the Vertex Cover problem once
more. As we observed already, a graph G with a maximum matching of size
µ has a minimum vertex cover of size β(G) = µ + k for some 0 ≤ k ≤ 2µ.
A 2-approximate algorithm for this problem simply includes all vertices of
a maximum matching. What is interesting is that no polynomial time algorithm is known for this problem which has an approximation factor a
constant strictly less than 2. In fact, it is now an outstanding open problem
whether there indeed exists such a polynomial time approximation algorithm.
An above-guarantee approximation algorithm for the Vertex Cover
problem tries to approximate k instead of the “entire” vertex cover. For
example, an algorithm which outputs a solution of size at most µ+α(β(G)−
µ), where α > 1, performs better than the 2-approximate algorithm (the one
which includes all vertices of a maximum matching) whenever µ + α(β(G) −
µ) < 2µ, that is, whenever β(G) − µ < µ/α. Since µ = O(n), this means
that whenever β(G) − µ < O(n/α), the additive approximation algorithm
beats the 2-approximate algorithm. Here n is the number of vertices in the
input graph.
Formally, one can define an above-guarantee α-approximate algorithm
as follows. If Q is an NP-maximization problem with a tight lower bound
(tlb) on its optimal solution size, then an above-guarantee α-approximate
algorithm for Q takes as input an instance I of Q and outputs a solution of
42
Chapter 2. Parameterizing From Default Values
size at least tlb(I) + α(opt(I) − tlb(I)) in time polynomial in |I|. For an
NP-minimization problem, an α-approximate algorithm outputs a solution
of size at most tlb(I) + α(opt(I) − tlb(I)) in time polynomial in |I|. Note
that for an NP-maximization problem α < 1; for a minimization problem
α > 1. One can now think of developing above-guarantee approximate
algorithms for other problems which have a guaranteed lower bound on
their solution size.
Open Problem 2.9. Does there exist an above-guarantee approximation
algorithm for the following problems?
1. Planar Independent Set.
2. Max Cut.
We note that the standard optimization versions of both these problems have good approximation algorithms. The Planar Independent
Set problem has a PTAS due to Baker [11]. For Max Cut, there exists a 0.879-approximate algorithm due to Goeman and Williamson [68].
Baker’s algorithm takes as input a planar graph G and a positive integer p
and outputs an independent set of size at least p/(p + 1) times optimal in
time O(8p p|V (G)|). Thus if a maximum independent set has size n/4 + k,
Baker’s algorithm outputs a solution of size at least p/(p+1)·(n/4+k). Here
again an above-guarantee approximate algorithm which outputs a solution
of size at least n/4 + αk, α a constant, yields a better solution whenever
n
p n
+ k < + αk,
p+1 4
4
that is, whenever
k<
n
.
4{p − α(p + 1)}
Thus if the optimum is only a “small distance” away from the lower bound,
an above-guarantee approximation performs better than the PTAS. A similar observation can be made for the Max Cut problem. This could be the
motivation for developing such algorithms.
In summary, we believe that parameterizing above or below guaranteed
bounds is an interesting paradigm both from a theoretical and practical
point of view with several important problems yet to be explored.
Chapter 3
Approximating Beyond the Limit of
Approximation
In the last chapter, we saw that if an NP-optimization problem Q admits
an α-approximation algorithm, then the α · opt ± k question is in FPT
provided the standard parameterized version of Q is in FPT. In this chapter we consider a related question, that of approximating beyond the limit
of approximation. The problem that we wish to address is this: Suppose
an NP-maximization problem admits a polynomial-time α-approximation
algorithm, where α < 1. What then is the complexity of finding an (α + ǫ)approximate solution given an arbitrary ǫ with 0 < ǫ < 1 − α? Since α could
be the limit of approximation for the problem, the algorithm in question
may require exponential time but the objective is to find a better-than-αapproximate solution much faster than it would take to compute an exact
solution. For minimization problems, we are interested in finding an (α − ǫ)approximate solution, where α > 1 and 0 < ǫ < α − 1.
Dantsin et al. [42] answer this question for Max Sat, Max 2-Sat
and Max 3-Sat. For Max Sat and Max 2-Sat, they give an algorithm
−1
that constructs √an (α + ǫ)-approximate solution in time O ∗ (φǫ(1−α) m),
where φ = (1 + 5)/2 is the golden ratio and m is the number of clauses in
the input formula. For Max 3-Sat, they construct an (α + ǫ)-approximate
−1
solution in time O ∗ (2ǫ(1−α) m ). We show that one can construct an (α + ǫ)approximate solution for a class of NP-maximization problems (and an
(α − ǫ)-approximate solution for NP-minimization problems) “efficiently”
provided that there exists an algorithm computing the optimum solution
that is “well-behaved” in some sense. We start with some basic definitions
regarding approximation algorithms.
3.1
Basic Definitions
Let Q = {I , S, V, opt} be an NP-optimization problem. For any instance I
of Q and any y ∈ S(I), the approximation ratio r(I, y) of y with respect
43
44
Chapter 3. Approximating Beyond the Limit of Approximation
to I is defined as
r(I, y) :=
V (I, y)
.
opt(I)
Therefore for maximization problems, the approximation ratio is a number ≤
1; for minimization problems, the ratio is ≥ 1. In either case, the closer the
ratio is to 1, the better the solution.
Definition 3.1. Let Q = {I , S, V, opt} be an NP-optimization problem
and let α > 0 be a real number.
1. A polynomial-time α-approximation algorithm for Q is a polynomialtime algorithm that, given an instance I ∈ I , computes a solution y ∈ S(I) such that r(I, y) ≥ α (if Q is a maximization problem)
and r(I, y) ≤ α (if Q is a minimization problem).
2. A polynomial time approximation scheme (PTAS) for Q is an algorithm A that takes as input a pair (I, k) ∈ I × N such that for
every fixed k, the algorithm is a polynomial-time approximation algorithm with ratio 1 + 1/k (if Q is a minimization problem) and with
ratio 1/(1 + 1/k) (if Q is a maximization problem).
Definition 3.2. An NP-optimization problem Q = {I , S, V, opt} is said to
be polynomially bounded if there exists a polynomial function b(·) such that
for all instances I ∈ I , we have opt(I) ≤ b(|I|). The function b(·) is called
the bound for the problem Q.
Max Sat, for instance, is polynomially bounded since the optimum
solution size is bounded above by the number of clauses in the input instance. Graph-theoretic optimization problems such as Vertex Cover,
Independent Set, Dominating Set, Feedback Vertex Set, Induced
Matching, where the solution is either a vertex or an edge subset, are other
examples of polynomially bounded NP-optimization problems. Depending
on whether the problem is a vertex or an edge problem, the bound is the
size of the vertex or edge set, respectively.
Definition 3.3. An NP-optimization problem Q is a subset-type optimization problem if for each instance I of Q there exists a set CI such that any
solution of I is a subset of CI .
For instance, Max Sat is a subset-type optimization problem since any
solution is simply a subset of the set of all clauses of the given input instance.
The other graph-theoretic problems mentioned above are subset-type problems too.
3.2. Related Work
3.2
45
Related Work
The idea of approximating a solution in exponential time is quite recent and
only a few papers have appeared on this topic [42, 40, 21]. This idea seems
to have originated in the paper by Dantsin et al. [42] where they show how
to construct an (α + ǫ)-approximate algorithm for Max Sat that runs in
−1
time O ∗ (φǫ(1−α)√ m ), given a polynomial-time α-approximation algorithm.
Here φ = (1 + 5)/2 is the “golden ratio” and m is the number of clauses
in the input formula. The motivation for developing such an algorithm
is that while Max Sat admits a 0.758-approximation algorithm [68], it
does not have a PTAS. In particular, there exists a constant α such that
a polynomial-time α-approximation algorithm for Max Sat implies P =
NP [8]. Therefore it is natural to investigate how fast one can obtain an
(α + ǫ)-approximate solution albeit in exponential time.
The approach used by Dantsin et al. can be described as follows. Many
exponential-time algorithms are recursive and can be viewed as visiting the
nodes of an exponential-sized search-tree. The nodes of the search-tree may
be viewed as tuples consisting of problem instances and partial solutions.
Typically the leaves correspond to either trivial instances or instances solvable in polynomial time. As the algorithm “descends” the search-tree, it
builds a solution whose size is typically proportional to the depth of recursion. The idea used by Dantsin et al. is to stop recursing as soon a partial
solution of sufficiently large size has been obtained and then applying the
approximation algorithm to the resulting instances. In this chapter we show
that not only can Max Sat, Max 2-Sat and Max 3-Sat be approximated
beyond the approximation lower bound using this technique, but so can
any subset-type optimization problem that admits an exponential-time algorithm with certain properties. In particular, we obtain (α+ǫ)-approximate
algorithms for Max Sat, Max 2-Sat and Max 3-Sat that run in the same
time as those obtained by Dantsin et al. in [42].
Another motivation for considering exponential-time approximation algorithms is that many problems such as Independent Set, Chromatic
Number, Subset Sum and a host of graph layout problems such as Bandwidth are hard to approximate in polynomial time (see [9] for more examples). For instance, Håstad [81] showed that Independent Set cannot be approximated in polynomial time to within n1−ǫ for any ǫ > 0 unless NP = ZPP, where n is the number of vertices in the input graph. Feige
and Killian [56] showed the same result for Chromatic Number. Feige [55]
also showed that Set Cover cannot be approximated in polynomial time
to within a factor (1 − ǫ) ln n, for any ǫ > 0, unless NP ⊆ DTIME(nlog log n ),
where n is the size of the set to be covered.
The parameterized versions of these problems are either not in FPT
unless P = NP or hard for various levels of the W-hierarchy [47]. Faced with
such a situation, it is natural to investigate whether one can obtain “good”
46
Chapter 3. Approximating Beyond the Limit of Approximation
(constant-factor) approximation algorithms for problems with running time
that is significantly better than the best-known exponential-time algorithms
for these problems. This is the approach adopted by Cygan et al. in [40] in
which they study two main techniques for developing such algorithms. The
first consists of designing a backtracking algorithm with a small search-tree
similar to the idea of Dantsin et al. [42]. Using this, they obtain a (4r − 1)approximation algorithm for Bandwidth in time O ∗ (2n/r ), for any r > 1.
The second technique is that of “reducing” an exponential-time algorithm to obtain approximate solutions so that the resulting algorithm runs
faster than the original algorithm (but still takes exponential time). We
illustrate this approach using Independent Set as an example. Given a
graph G = (V, E), arbitrarily partition the vertex set V into r sets V1 , . . . , Vr
of (almost) equal size. Now find a maximum independent set Ij in each
of the graphs G[Vj ], 1 ≤ j ≤ r, using a brute-force algorithm that runs
through all vertex subsets and checks if they are independent. Clearly the
largest Ij is an r-approximate solution for G and the time taken to obtain such a set is O ∗ (2n/r ), which is better than a running time of O∗ (2n )
taken by the algorithm to obtain an exact solution. Cygan et al. refine
this idea and obtain a (1 + ln r)-approximation algorithm for Set Cover in
time O ∗ (2n/0.31r ); and an r-approximation algorithm for Weighted Dominating Set in time O∗ (20.589n/r ), for any r > 1.
Bourgeois et al. [21] use ideas similar to that of “reductions” described
in the previous paragraph to show, among other things, that a problem
called Max Hereditary-π can be approximated to any factor 0 < α < 1
in moderately exponential time. This problem is defined as follows: given
a graph G = (V, E) and some hereditary property π, find a subset V ′ ⊆ V
of maximum size such that G[V ′ ] satisfies property π. Bourgeois et al.
show that given an exact algorithm for Max Hereditary-π with worstcase complexity O ∗ (γ n ), and a rational 0 < α < 1, there exists an αapproximation algorithm for the problem with running time O∗ (γ αn ). Using this result and the analysis of a linear program for the Vertex Cover
problem by Nemhauser and Trotter [106], they show that given an exact
algorithm for Independent Set with running time O ∗ (γ n ) and a rational 0 < ǫ < 1, one can obtain a (2 − ǫ)-approximate solution for Vertex
Cover in time O ∗ (γ ǫn ).
Vassilevska et al. [122] study an interesting variation of the approximation versus exact algorithms theme called hybrid algorithms. A hybrid algorithm for an NP-optimization problem Q is a collection of algorithms H = {h1 , . . . , hp } called heuristics, coupled with a polynomial-time
procedure S called the selector. Given an instance I of Q, the selector returns an index i, 1 ≤ i ≤ p, and then the heuristic hi is executed on I.
The purpose of the selector is to select the most appropriate heuristic for a
given instance. Vassilevska et al. focus on hybrids consisting of two heuristics: the first is a super-polynomial time exact algorithm and the second is
3.3. Approximating Beyond Approximation Limits
47
a polynomial-time approximation algorithm. They go on to show that the
Max Cut, Longest Path, Bandwidth and Constraint Satisfaction
problems admit hybrid algorithms where a given instance is solved exactly
in time better than the best-known exact algorithm for the problem or is
approximated in polynomial time to within a ratio exceeding the known
inapproximability of the problem.
3.3
Approximating Beyond Approximation Limits
In what follows, we will assume that the NP-optimization problems we consider are subset-type problems and that given an instance I of such a problem, C(I) represents the finite set that contains all possible solutions to I.
We also let Opt(I) denote an optimum solution to the instance I (and we
let opt(I) represent the size of an optimum solution). We begin by formalizing what we mean by an NP-optimization problem admitting a “wellbehaved” exact algorithm.
Definition 3.4. An algorithm A is an (f, g)-exact algorithm for an NPoptimization problem Q = {I , S, V, opt} if it satisfies the following property: A takes as instance a tuple (I, r) ∈ I × N and in at most f (r, |I|)
steps
1. either produces an optimum solution if opt(I) ≤ r, or
2. produces at most g(r) tuples (I1 , T1 ), . . . , (Ig(r) , Tg(r) ), where Ti ⊆
C(I) \ C(Ii ) and |Ti | ≥ r, for all 1 ≤ i ≤ r, and each Ii is an instance of Q such that for some 1 ≤ j ≤ g(r), Opt(Ij ) ∪ Tj = Opt(I).
Notice that the conditions Ti ⊆ C(I)\C(Ii ) and Opt(Ij )∪Tj = Opt(I),
imply that opt(Ij ) + |Tj | = opt(I), for some index j. Before proceeding
any further let us look at some examples of such exact algorithms. In what
follows we let n and m denote the number of vertices and edges, respectively,
of a graph, unless otherwise stated.
Example 3.1. Vertex Cover. The following algorithm, that takes a
graph G = (V, E) and a positive integer r as input, is an (f, g)-exact algorithm for Vertex Cover, where f (r, |G|) = 2r+1 (n + m) and g(r) = 2r .
The algorithm picks an arbitrary edge {u, v} in the graph that has not yet
been covered and considers two cases: either u is in the solution or else v is
in the solution. In the first case, the algorithm recurses on the graph G − u
after taking u in the solution and in the second it recurses on G − v after
taking v in the solution.
The recursion tree may be viewed as a rooted binary tree whose nodes
are labelled by pairs of the form (Gi , Ti , l), where Gi is a subgraph of G,
Ti ⊆ V (G) and l denotes the depth of node from the root. The root is
48
Chapter 3. Approximating Beyond the Limit of Approximation
labelled (G, ∅, 0). The algorithm recurses till depth at most r; if it finds a
vertex cover it outputs it, or it outputs the 2r tuples (Gi , Ti , r) corresponding
to the leaves of the recursion tree. Note that partial solutions corresponding
to nodes at level i are of size i. It is easy to verify that if r ≥ opt(G) then
the algorithm finds an optimal solution; otherwise for some 1 ≤ j ≤ 2r ,
there exists a node (Gi , Ti , r) such that Opt(Gi ) ∪ Ti = Opt(G).
The next example that we consider is the Independent Set problem in
graphs of bounded degeneracy. A graph is said to be d-degenerate if every
subgraph of it has a vertex of degree at most d [125]. Examples of graphs of
bounded degeneracy include planar graphs (which are 5-degenerate), graphs
of bounded degree, graphs of bounded genus, Kh -minor-free graphs, Kh topological-minor-free graphs, for every fixed positive integer h.
Example 3.2. The Independent Set problem in graphs of degeneracy d.
An (f, g)-exact algorithm for this problem with f (r, |G|) = (d+1)r+1 (n+m)
and g(r) = (d+1)r is as follows. Given a d-degenerate graph G = (V, E) and
a positive integer r, the algorithm picks a vertex u of degree at most d and
considers at most d + 1 cases. In each case, the algorithm includes exactly
one vertex from N [u] in the solution, deletes its closed neighborhood from
the graph, and recurses on the new instance thus created. The recursion
tree is now a (d + 1)-ary tree and the algorithm recurses till depth at most r.
Again it is easy to see that if r ≥ opt(G) then the algorithm finds an
optimal solution, or else outputs the (d + 1)r instance-partial-solution pairs
corresponding to the leaves of the tree such that property 2 of Definition 3.4
holds.
Example 3.3. Next consider the Dominating Set problem in graphs of
maximum degree at most B, a constant. Given such a graph G = (V, E) and
a positive integer r, our algorithm first colors all vertices of G red. It picks
a vertex u and considers B + 1 cases and, in each case, includes exactly one
vertex v ∈ N [u] in the dominating set, colors all neighbors of v blue, and
deletes v from the graph. Intuitively, red vertices are those that are yet to be
dominated and the blue ones are those that have already been dominated. If
a blue vertex has only blue neighbors then it is deleted from the graph as it
plays no role in dominating red vertices. The algorithm recurses till depth r
and either ends up with a graph with only blue vertices or with (B + 1)r
instance-partial-solution pairs satisfying property 2 of Definition 3.4. The
time taken is (B + 1)r+1 (n + m).
Example 3.4. We also note that Mahajan and Raman [98] describe an
(f, g)-exact algorithm for Max Sat which takes a Boolean Cnf formula F
and a positive integer r as input, and in time
f (r, |F |) = O r 2 · φr + |F |
3.3. Approximating Beyond Approximation Limits
49
either outputs an optimum solution or g(r) = φr instance-partial-solution
√
pairs such that property 2 of Definition 3.4 holds [98]. Here φ = (1 + 5)/2.
Our first observation regarding (f, g)-exact algorithms is that if the function f is “FPT-like” then the standard parameterized version of the problem
in question is fixed-parameter tractable.
Theorem 3.1. If an NP-optimization problem Q admits an (f, g)-exact
algorithm such that f (r, n) = h(r) · p(n), where h is a function of r alone
and p is a polynomial, then the standard parameterized version of Q is fixedparameter tractable.
Proof. Suppose Q is a maximization problem and let A be an (f, g)-exact
algorithm for it. Given an instance (I, k) of the standard parameterized
version of Q, run algorithm A on I for at most f (k−1, |I|) steps. If it outputs
a solution then it must be that opt(I) ≤ k − 1 and we answer no. Else
answer yes. The case when Q is a minimization problem can be similarly
dealt with.
We now come to the main result of this chapter.
Theorem 3.2. Let Q be a polynomially-bounded NP-maximization problem
with bound b(·) that satisfies the following properties:
1. Q admits an α-approximation algorithm that runs in time p(n), for
some 0 < α < 1 and some polynomial p;
2. Q admits an (f, g)-exact algorithm.
Then given 0 < ǫ < 1 − α and an instance I of Q of size n, one can compute
an (α + ǫ)-approximate solution in time
ǫ · b(n)
ǫ · b(n)
f
,n + g
· p(n).
1−α
1−α
Proof. Let A be an (f, g)-exact algorithm for Q. Given an instance I of
size n, run A on it for f (cǫ,n , n) steps, where
cǫ,n =
ǫ · b(n)
.
1−α
If the algorithm outputs an optimum solution within this time, then this
solution is trivially (α + ǫ)-approximate. Otherwise it outputs g(cǫ,n ) tuples (Ii , Ti ). Next run the polynomial-time α-approximation algorithm for Q
on the instances Ii , 1 ≤ i ≤ g(cǫ,n ), and let Ai represent the solution thus
obtained for instance Ii . Let i be an index for which |Ai ∪ Ti | is maximized. We claim that Ai ∪ Ti is an (α + ǫ)-approximate solution for I.
By the definition of an (f, g)-exact algorithm, there exists 1 ≤ j ≤ g(cǫ,n )
50
Chapter 3. Approximating Beyond the Limit of Approximation
such that opt(I) = |Tj | + opt(Ij ). Also |Tj | ≥ cǫ,n and |Aj | ≥ α · opt(Ij ).
Therefore,
|Aj | + |Tj | ≥ α · opt(Ij ) + |Tj |
≥ α · opt(I) + (1 − α) · |Tj |
≥ α · opt(I) + ǫ · b(n)
≥ (α + ǫ) · opt(I).
It is easy to see that the running time is as claimed in the statement of the
theorem.
Since Max Sat has an (f, g)-exact
algorithm with f (r, m) = O(r 2 φr +m)
√
r
and g(r) = φ , where φ = (1 + 5)/2, by Theorem 3.2 we have
Corollary 3.1. Assume that there exists an α-approximation algorithm for
Max Sat that runs in time p(m), where m is the number of clauses in
the input formula F . Then for any 0 < ǫ < 1 − α, one can construct an
(α + ǫ)-approximate solution for Max Sat in time O((c2ǫ,m + p(m))φcǫ,m ),
√
where cǫ,m = ǫ · (1 − α)−1 · m and φ = (1 + 5)/2.
The running time bound obtained in Corollary 3.1 is the same as that
obtained by Dantsin et al. in [42].
By Example 3.2, the Independent Set problem in d-degenerate graphs
has an (f, g)-exact algorithm where f (r, |G|) = (d+1)r+1 ·(n+m) and g(r) =
(d + 1)r . By Theorem 3.2 we have
Corollary 3.2. If the Independent Set problem in d-degenerate graphs
admits an α-approximation algorithm that runs in time p(n), where n is the
number of vertices in the input graph, then given any 0 < ǫ < 1 − α, one
can construct an (α + ǫ)-approximate solution in time O((d + 1)cǫ,n · p(n)),
where cǫ,n = ǫn/(1 − α).
For minimization problems, we have the following theorem.
Theorem 3.3. Let Q be a polynomially-bounded NP-minimization problem
with bound b(·) that satisfies the following properties:
1. Q admits an α-approximation algorithm that runs in time p(n), for
some α > 1 and some polynomial p;
2. Q admits an (f, g)-exact algorithm.
Then given 0 < ǫ < α − 1 and an instance I of Q of size n, one can compute
an (α − ǫ)-approximate solution in time
ǫ · b(n)
ǫ · b(n)
,n + g
· p(n).
f
α−1
α−1
3.3. Approximating Beyond Approximation Limits
51
Proof. This proof of this theorem is similar to the one for Theorem 3.2 but
we include it for the sake of completeness. Let A be an (f, g)-exact algorithm
for Q. Given an instance I of size n, run A on I for f (cǫ,n , n) steps, where
cǫ,n =
ǫ · b(n)
.
α−1
If the algorithm outputs an optimum solution then this is trivially an (α−ǫ)approximate solution. Otherwise A outputs g(cǫ,n ) tuples (Ii , Ti ). Next run
the polynomial-time α-approximation algorithm for Q on the instances Ii ,
1 ≤ i ≤ g(cǫ,n ), and let Ai represent the solution obtained by running this
algorithm on Ii . Let i be an index for which |Ai ∪ Ti | is maximized. We
claim that Ai ∪ Ti is an (α − ǫ)-approximate solution for the instance I.
By the definition of an (f, g)-exact algorithm, there exists 1 ≤ j ≤ g(cǫ,n )
such that opt(I) = |Tj | + opt(Ij ). Also |Tj | ≥ cǫ,n and |Aj | ≤ α · opt(Ij ).
Therefore,
|Aj | + |Tj | ≤ α · opt(Ij ) + |Tj |
≤ α · opt(I) − (α − 1) · |Tj |
≤ α · opt(I) − ǫ · b(n)
≤ (α − ǫ) · opt(I).
It is easy to see that the running time is as claimed in the statement of the
theorem.
Since Vertex Cover admits a 2-approximation algorithm that runs in
time O(m) [38] and an (f, g)-exact algorithm with f (r, |G|) = 2r+1 (n + m)
and g(r) = 2r , we have
Corollary 3.3. For every 0 < ǫ < 1, there exists an approximation algorithm for the Vertex Cover problem with ratio 2 − ǫ that runs in
time O(2ǫn · m).
Note that this running time is worse than that obtained by Bourgeois et
al. [21] for a (2 − ǫ)-approximation algorithm for Vertex Cover, but our
results are more general and have wider applicability.
For the Dominating Set problem in graphs with degree bounded by B,
there exists an (f, g)-exact algorithm with f (r, |G|) = (B + 1)r+1 (n + m)
and g(r) = (B + 1)r .
Corollary 3.4. Suppose that the Dominating Set problem in graphs with
degree bounded by B, admits an α-approximation algorithm that runs in
time p(n). Then given any 0 < ǫ < α − 1, there exists an approximation
algorithm for this problem with ratio α − ǫ that runs in time O((B + 1)cǫ,n ·
p(n)), where cǫ,n = ǫn/(α − 1).
52
Chapter 3. Approximating Beyond the Limit of Approximation
We mention one final result of this type. The Almost d-Regular Induced Subgraph problem is defined as follows: given a graph G, what
is the fewest number of vertices that need to be deleted so that the resulting graph has maximum degree d? In [103], Moser and Thilikos give an
(f, g)-exact algorithm for this problem with f (r, |G|) = (d + 1)r+1 (n + m)
and g(r) = (d + 1)r . We therefore obtain
Corollary 3.5. If the Almost d-Regular Induced Subgraph problem
admits an α-approximation algorithm that runs in time p(n), then given
any 0 < ǫ < α − 1, there exists an approximation algorithm for this problem
with ratio α−ǫ that runs in time O((d+1)cǫ,n ·p(n)), where cǫ,n = ǫn/(α−1).
3.4
Concluding Remarks
In this chapter we considered designing (exponential-time) algorithms with
an approximation ratio that is a fraction ǫ away from a ratio that can be
achieved in practice. For small enough ǫ, these algorithms take moderately
exponential time and are much faster than exponential-time algorithms that
compute optimum solutions. This line of research is still in its infancy and
there are quite a few open questions.
A natural question is whether one can obtain results similar to Theorems 3.2 and 3.3 by relaxing the notion of an (f, g)-exact algorithm. Are
there any connections with parameterized enumerability? Another question
is to design algorithms for specific problems such as Feedback Vertex
Set. This problem admits a 2-approximation algorithm due to Bafna et
al. [10] and admits an O ∗ (1.755n ) exact algorithm due to Gaspers et al. [60].
How quickly can one obtain a (2−ǫ)-approximate solution? What about Directed Feedback Vertex Set which admits an O(log n log log n)-ratio
algorithm [53]?
Another interesting problem is Max Cut which admits exact algorithms with running times O∗ (2ωn/3 ) [126], O ∗ (2m/4 ) [54], and O ∗ (2m/5 ) [92],
where n and m denote the number of vertices and edges of the input graph,
respectively. This problem also admits a 0.879-approximation algorithm due
to the seminal work by Goemans and Williamson [68]. Vassilevska et al. [122]
give a hybrid algorithm for Max Cut that given an ǫ > 0, either computes
an exact solution in time 2ǫm or finds a (1/2 + ǫ/4)-approximate solution in
linear time. But how fast can one obtain an (0.879+ǫ)-approximate solution
for Max Cut?
In summary, we believe that the issues considered in this chapter deserve
to be explored further and that there are many interesting questions worth
investigating.
Chapter 4
Kőnig Subgraph Problems and
Above-Guarantee Vertex Cover
In this chapter we study the complexity of a set of problems which we call
Kőnig Subgraph Problems. These problems consist of finding subgraphs
of a given graph with the property that the size of a minimum vertex cover
equals that of a maximum matching in the subgraph. Graphs that satisfy
this property are called Kőnig-Egerváry graphs. More specifically, we look
at the following two sets of problems. Given a graph G and a nonnegative
integer k,
1. does there exist k vertices (edges) whose deletion makes the graph
Kőnig-Egerváry?
2. does there exist k vertices (edges) that induce a Kőnig-Egerváry subgraph?
We show that these problems are NP-complete and study their complexity
from the points of view of approximation and parameterized complexity.
While this may seem to be a marked departure from the study of parameterizing problems from their default values, we will see that the vertex
deletion version is closely related to Above Guarantee Vertex Cover
(see Chapters 1 and 2). In our study of the parameterized complexity of
the vertex deletion version, we uncover a number of interesting structural
results on matchings and vertex covers which may be independent interest.
4.1
History and Motivation
The classical notions of matchings and vertex covers have been at the center
of serious study for several decades in the area of Combinatorial Optimization [97]. In 1931, Kőnig and Egerváry independently proved a result of fundamental importance: in a bipartite graph the size of a maximum matching
equals that of a minimum vertex cover [97]. This led to a polynomial-time
53
54
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
algorithm for finding a minimum vertex cover in bipartite graphs. In fact, a
maximum matching can be used to obtain a 2-approximation algorithm for
the Minimum Vertex Cover problem in general graphs, which is still the
best-known constant-factor approximation algorithm for this problem [87].
Interestingly, this min-max relationship holds for a larger class of graphs
known as Kőnig-Egerváry graphs and it includes bipartite graphs as a proper
subclass. Kőnig-Egerváry graphs will henceforth be called Kőnig graphs.
Kőnig graphs have been studied for a fairly long time from a structural
point of view [22, 44, 50, 96, 121]. Both Deming [44] and Sterboul [121]
gave independent characterizations of Kőnig graphs and showed that Kőnig
graphs can be recognized in polynomial time. Lovász [96] used the theory of
matching-covered graphs to give an excluded-subgraph characterization of
Kőnig graphs that contain a perfect matching. Korach et al. [50] generalized
this and gave an excluded-subgraph characterization for the class of all Kőnig
graphs.
A natural optimization problem associated with a graph class G is the
following: given a graph G, what is the minimum number of vertices to be
deleted from G to obtain a graph in G? For example, when G is the class of
empty graphs, forests or bipartite graphs, the corresponding problems are
Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal, respectively. We call the vertex-deletion problem corresponding to the
class of Kőnig graphs the Kőnig Vertex Deletion problem. A set of
vertices whose deletion makes a given graph Kőnig is called a Kőnig vertex
deletion set. In the parameterized setting, the parameter for vertex-deletion
problems is the solution size, that is, the number of vertices to be deleted
so that the resulting graph belongs to the given graph class.
In this chapter we define various problems related to finding KőnigEgerváry subgraphs and study their complexity from the points of view of
parameterized complexity and approximation algorithms. More precisely
the problems that we study in this chapter are:
1. Kőnig Vertex (Edge) Deletion (KVD/KED). Given a graph G
and a nonnegative integer k, do there exist at most k vertices (respectively, edges) whose deletion results in a Kőnig subgraph?
2. Vertex (Edge) Induced Kőnig Subgraph (VKS/EKS). Given a
graph G and a nonnegative integer k, do there exist at least k vertices
(respectively, edges) which induce1 a Kőnig subgraph?
The KVD and VKS problems (and similarly, KED and EKS) are equivalent
from the point of view of NP-completeness but differ in their approximability
and parameterized complexity.
1
If E ′ is an edge-subset, the graph G[E ′ ] induced by E ′ is defined as one with vertex
set V (E ′ ) and edge-set E ′ .
4.2. Preliminaries
55
While studying the Kőnig Vertex Deletion problem, we first establish interesting structural connections between minimum vertex covers,
maximum matchings and minimum Kőnig vertex deletion sets. Using these,
we show that Kőnig Vertex Deletion is fixed-parameter tractable when
parameterized by the solution size. Note that Kőnig graphs are not hereditary, that is, not closed under taking induced subgraphs. For instance,
a 3-cycle is not Kőnig but attaching an edge to one of the vertices of the
3-cycle results in a Kőnig graph. In fact, Kőnig Vertex Deletion is one
of the few vertex-deletion problems associated with a non-hereditary graph
class whose parameterized complexity has been studied. Another such example can be found in [103].
One motivation for studying Kőnig subgraph problems is that the versions of Kőnig subgraph problems when the resulting graph we look for is
bipartite (i.e. replace Kőnig in the above problem definitions by bipartite)
are well studied in the area of approximation algorithms and parameterized
complexity [70, 117, 120]. Kőnig subgraph problems are natural generalizations of bipartite subgraph problems but have not been studied algorithmically. We believe that this can trigger explorations of other questions in
Kőnig graphs. Another motivation for studying Kőnig subgraph problems is
that Kőnig Vertex Deletion is closely related to Above Guarantee
Vertex Cover. As mentioned in Chapter 2, the parameterized complexity
of Above Guarantee Vertex Cover was open for quite some time and
is now known to be fixed-parameter tractable as we show in this chapter.
The rest of this chapter is organized as follows. In Section 4.2, we describe our notation and state some known results about Kőnig graphs. In
Section 4.3 we show that Above Guarantee Vertex Cover is fixedparameter tractable and discuss its approximability. We next study the
parameterized complexity and approximability of Kőnig Vertex Deletion (Section 4.4) and the vertex and edge versions of the Induced Kőnig
Subgraph problem (Section 4.5). We conclude in Section 4.6 with a list
of open problems. Missing from our list is Kőnig Edge Deletion the
parameterized complexity of which is open.
4.2
Preliminaries
In this section we fix our notation and describe some well-known properties
of Kőnig graphs.
4.2.1
Notation
Given a graph G, we use µ(G), β(G) and κ(G) to denote, respectively,
the size of a maximum matching, a minimum vertex cover and a minimum
Kőnig vertex deletion set of G. We sometimes use τ (G) to denote the difference β(G) − µ(G). When the graph being referred to is clear from the con-
56
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
text, we simply use µ, β, κ and τ . Given a graph G = (V, E) and two disjoint
vertex subsets V1 , V2 of V , we let (V1 , V2 ) denote the bipartite graph with
vertex set V1 ∪ V2 and edge set {{u, v} : {u, v} ∈ E and u ∈ V1 , v ∈ V2 }. If B
is a bipartite graph with vertex partition L ⊎ R then we let µ(L, R) denote
the size of the maximum matching of B. If M is matching and {u, v} ∈ M
then we say that u is the partner of v in M . If the matching being referred
to is clear from the context we simply say u is a partner of v. The vertices of G that are the endpoints of edges in the matching M are said to be
saturated by M ; all other vertices are unsaturated by M .
4.2.2
Properties of Kőnig Graphs
A graph G = (V, E) is said to be Kőnig if β(G) = µ(G). The following
lemma follows directly from the definition of Kőnig graphs.
Lemma 4.1. [44, 121] A graph G = (V, E) is Kőnig if and only if for every
bipartition of V into V1 ⊎ V2 , with V1 a minimum vertex cover of G, there
exists a matching across the cut (V1 , V2 ) saturating every vertex of V1 .
In order to show that a graph is Kőnig it is actually sufficient to demonstrate the existence of just one bipartition of V into V1 ⊎ V2 , with V1 a
vertex cover of G such that there exists a matching across the cut (V1 , V2 )
saturating every vertex of V1 .
Lemma 4.2. A graph G = (V, E) is Kőnig if and only if there exists a
bipartition of V into V1 ⊎ V2 , with V1 a vertex cover of G such that there
exists a matching across the cut (V1 , V2 ) saturating every vertex of V1 .
Proof. If G is Kőnig then, by Lemma 4.1, there exists a bipartition V1 ⊎ V2 ,
with V1 a minimum vertex cover of G, such that there exists a matching
across the cut (V1 , V2 ) saturating every vertex of V1 . Conversely suppose
that the vertex set of G can be partitioned as V1 ⊎ V2 such that V1 is a
vertex cover and there exists a matching M across the cut (V1 , V2 ) saturating
every vertex of V1 . We claim that in fact V1 is a minimum vertex cover and
that M is a maximum matching of G. Suppose that M ′ is a maximum
matching of G and |M ′ | > |M |. Since V1 is a vertex cover, it picks up at
least one endpoint from each edge of M ′ . Therefore |V1 | = |M | ≥ |M ′ |, a
contradiction. Therefore M is indeed a maximum matching of G and since
any vertex cover of G has size at least |M |, it follows that V1 is a minimum
vertex cover of G.
Lemma 4.3. [44] Given a graph G on n vertices and m edges and a maximum matching of G, one can test whether G is Kőnig in time O(n + m).
If G is indeed Kőnig then one can find a minimum vertex cover of G in this
time.
4.3. The Above Guarantee Vertex Cover Problem
57
√
Since a maximum matching can be obtained in time O(m n) [123], we
have
Lemma 4.4. Let G be a graph on n vertices and m edges. One can check in
√
time O(m n) whether G is Kőnig and, if Kőnig, find a bipartition of V (G)
into V1 ⊎ V2 with V1 a minimum vertex cover of G such that there exists a
matching across the cut (V1 , V2 ) saturating every vertex of V1 .
4.3
The Above Guarantee Vertex Cover Problem
In this section we show that Above Guarantee Vertex Cover (AGVC)
is fixed-parameter tractable and discuss its approximability. This problem
plays a central role in this chapter and the results established here are used in
studying the parameterized complexity and approximability of other Kőnig
subgraph problems. We will show later (Theorem 4.7) that for the class
of graphs with a perfect matching, the parameterized complexity of AGVC
and Kőnig Vertex Deletion is the same. For graphs G that do not have
a perfect matching, there is a close relationship between τ (G) and κ(G).
Given a graph G it is clear that β(G) ≥ µ(G). Recall the definition
of AGVC: given a graph G and a nonnegative integer parameter k decide
whether β(G) ≤ µ(G) + k. We first show that for the parameterized complexity of the AGVC problem we may, without loss of generality, assume
that the input graph has a perfect matching.
Let G = (V, E) be an undirected graph and let M be a maximum matching of G. Construct G′ = (V ′ , E ′ ) as follows. Define
I = V \ V [M ]
V ′ = V ∪ {u′ : u ∈ I}
E ′ = E ∪ {{u′ , v} : {u, v} ∈ E} ∪ {{u, u′ } : u ∈ I}.
Then M ′ = M ∪ {{u, u′ } : u ∈ I} is a perfect matching for G′ . Note
that |V (G′ )| ≤ 2|V (G)| and |E(G′ )| ≤ 2|E(G)|.
Theorem 4.1. Let G be a graph without a perfect matching and let G′ be
the graph obtained by the above construction. Then G has a vertex cover of
size µ(G) + k if and only if G′ has a vertex cover of size µ(G′ ) + k.
Proof. Let M denote a maximum matching of G, I denote the set V (G) \
V [M ] and I ′ denote the new set of vertices that are added in constructing
G′ . Clearly, µ(G′ ) = µ(G) + |I|.
(⇒) Let C be a vertex cover of G of size µ(G) + k. Define C ′ = C ∪ I ′ . It
is easy to see that C ′ covers all the edges of G′ . Also, |C ′ | = µ(G)+k +|I ′ | =
µ(G′ ) + k.
(⇐) Let C ′ be a vertex cover of G′ of size µ(G′ ) + k. Define M ′ to be the
set of edges of the form {{u, u′ } : u ∈ I and u′ ∈ I ′ } such that both endpoints
58
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
are in C ′ . One can show that C = (C ′ ∩ V [M ]) ∪ {u ∈ I : {u, u′ } ∈ M ′ } is a
vertex cover of G of size µ(G) + k.
4.3.1
Parameterized Complexity
We show that AGVC is fixed-parameter tractable by exhibiting a fixedparameter reduction from AGVC to a problem know as Min 2-Cnf Sat
Del [98]. This problem is defined as follows: given a 2-Cnf formula and a
nonnegative integer k, do there exist at most k clauses whose deletion makes
the resulting formula satisfiable? This problem is NP-complete and its parameterized complexity was open for quite some time. Recently Razgon and
O’Sullivan have shown this problem to be fixed-parameter tractable [116].
Theorem 4.2. [116] Given a 2-Cnf Sat formula F on n variables and m
clauses and a nonnegative integer k, one can decide whether F has at most k
clauses whose deletion makes it satisfiable in time O(15k · k · m3 ). That is,
the Min 2-Cnf Sat Del problem is fixed-parameter tractable with respect
to parameter k.
We now describe the reduction from AGVC to Min 2-Cnf Sat Del
(see [29]). Let G = (V, E) be a graph with a perfect matching P . For every
vertex u ∈ V , define xu to be a Boolean variable. Let F(G, P ) denote the
Boolean formula
^
^
F(G, P ) =
(x̄u ∨ x̄v )
(xu ∨ xv ).
(u,v)∈P
(u,v)∈E
Note that F(G, P ) is a formula on |V | variables and at most 2|E| clauses.
The proof of the next lemma follows from that of Theorem 5.1 in [29].
Lemma 4.5. Let G = (V, E) be an n-vertex graph with a perfect matching P . Then G has a vertex cover of size at most n/2 + k if and only if there
exists an assignment that satisfies all but at most k clauses of F(G, P ).
From the proof of Theorem 5.1 in [29], it also follows that given an
assignment that satisfies all but at most k clauses of F(G, P ) one can find
(in polynomial time) an assignment that satisfies all but at most k clauses
of the form (x̄u ∨ x̄v ), where (u, v) ∈ P , that is, clauses that correspond to
the perfect matching.
Since Min 2-Cnf Sat Del can be solved in time O(15k ·k·m3 ), where m
is the number of clauses in the input formula, we have
Theorem 4.3. Given a graph G = (V, E) and a nonnegative integer parameter k, one can decide whether β(G) ≤ µ(G) + k in time O(15k · k · |E|3 ).
Moreover if G has a vertex cover of size µ(G) + k then one can find a vertex
cover of this size within this time.
4.3. The Above Guarantee Vertex Cover Problem
4.3.2
59
An Approximation Algorithm
The parameterized version of AGVC asks whether τ (G) ≤ k. The optimization version of AGVC is the problem of finding the minimum value
of τ (G). Therefore an approximation algorithm for AGVC approximates
the “above-guarantee parameter” rather than the entire vertex cover.
Klein et al. [90] have shown that Min 2-Cnf Sat Del admits a factorO(log n log log n) approximation algorithm, where n is the number of variables in the 2-Sat formula.
Lemma 4.6. [1, 90] Given an instance F of Min 2-Cnf Sat Del, one can
obtain a solution for F in polynomial time that is O(log n log log n) times
the optimal solution size, where n is the number of variables in F √
. If we
are willing to allow randomness, we can obtain a solution that is O( log n)
times an optimal solution size.
We use this algorithm and the reduction from AGVC to Min 2-Cnf Sat
Del to obtain an O(log n log log n)-approximation algorithm for τ (G).
Here is an outline of our approximation algorithm: Given a graph G,
first apply the construction described before Theorem 4.1, if necessary, to
obtain a graph H with a perfect matching P . Note that τ (G) = τ (H).
Let F(H, P ) denote the 2-Cnf Sat formula obtained from H and P by the
construction outlined before Lemma 4.5. Given an assignment that satisfies
all but at most k clauses of F(H, P ) one can construct an assignment in
polynomial time that satisfies all but at most k clauses of the form (x̄u ∨ x̄v ),
where (u, v) ∈ P .
Next use an O(log n log log n) approximation algorithm for Min 2-Cnf
Sat Del which “corresponds” to a set S of edges of the perfect matching P .
The set V (S) represents the vertex cover in excess of the matching size and
in the graph G \ V (S), the sizes of a minimum vertex cover and maximum
matching coincide. That is, G \ V (S) is Kőnig and therefore by Lemma 4.4
one can obtain a minimum vertex cover C of this graph in polynomial time.
Using C and S, reconstruct a vertex cover for G of the appropriate size.
The algorithm is presented in Figure 4.1.
Theorem 4.4. Given an input graph G on n vertices, algorithm AGVC
finds a vertex cover of G of size µ + O(log n log log n)(β − µ).
Proof. The proof follows from the fact that the reduction from AGVC to
Min 2-Cnf Sat Del is cost-preserving and that there exists a factorO(log n log log n) approximation algorithm for the latter.
Thus this algorithm approximates the deficit between the sizes of a minimum vertex cover and a maximum matching. We mentioned that there
exists a 2-approximation algorithm for the Vertex Cover problem and
that it is a long-standing open problem to devise a polynomial-time algorithm which has a constant approximation factor less than 2.
60
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
AGVC
Input: A graph G = (V, E).
Output: A vertex cover of G of size at most µ + O(log |V | log log |V |)(β − µ).
1. If G does not have a perfect matching, construct G′ as in Theorem 4.1 and
set H ← G′ ; else set H ← G.
2. Find a perfect matching P of H and construct F (H, P ). If G did not have a
perfect matching then P is the perfect matching obtained from some maximum matching M of G as described in the construction before Theorem 4.1.
3. Use the approximation algorithm for Min 2-Cnf Sat Del to obtain an
O(log n log log n)-approximate solution S for F (H, P ), where n = |V (H)|.
4. Obtain a minimum vertex cover C of the Kőnig graph H \ V (S), where V (S)
is the set of vertices of H corresponding to S.
5. If H = G then return C ∪ V (S); else return (V (S) ∩ V (G)) ∪ (V (M ) ∩ C).
Figure 4.1: An approximation algorithm for AGVC.
Our algorithm is better than any constant factor approximation algorithm for Vertex Cover whenever
n
β−µ=o
log n log log n
and µ = Ω(n). To see this, note that a c-approximate algorithm, c > 1,
outputs a solution of size µc + (β − µ)c whereas our algorithm outputs a
solution of size µ + O(α(β − µ)), where α = log n log log n. Now if β −
µ = o(n/α) and if µ = Ω(n), then our algorithm outputs a solution of
size µ + o(µ), which is better than
βc = µ + µ(c − 1) + (β − µ)c ≥ µ + Ω(µ).
A randomized
approximation algorithm for τ can be obtained by using
√
the O( log n)-randomized approximation algorithm for Min 2-Cnf Sat
Del [1], mentioned in Lemma 4.6, in Step 3 of the algorithm.
Theorem 4.5. There exists a randomized polynomial-time algorithm that
takes as√input a graph G on n vertices and finds a vertex cover of G of size
µ + O( log n)(β − µ).
4.3.3
Hardness of Approximation
We now show that Above Guarantee Vertex Cover and Min 2-Cnf
Sat Del do not admit constant-factor approximation algorithms if the
Unique Games Conjecture (UGC) [85] is true2 . In what follows, we
2
I thank Sundar Vishwanathan for discussions on the results in this section, and in
particular, Theorem 4.6.
4.3. The Above Guarantee Vertex Cover Problem
61
use the abbreviation VC-PM for the Vertex Cover problem on graphs
with a perfect matching.
We make use of the following two results:
Lemma 4.7. [87] If UGC is true then Vertex Cover cannot be approximated to within a factor of 2 − ǫ, for any constant ǫ > 0.
Lemma 4.8. [29, 124] If there exists a (2 − ǫ)-approximation algorithm for
VC-PM then there exists a (2 − ǫ/2)-approximation algorithm for Vertex
Cover.
We can now prove the following.
Theorem 4.6. Assuming UGC to be true, the Above Guarantee Vertex Cover problem in graphs with a perfect matching cannot be approximated to within a factor of c, for any constant c > 1.
Proof. Suppose that there exists a c-approximate algorithm A for Above
Guarantee Vertex Cover on graphs with a perfect matching for some
constant c > 1. By Lemmas 4.7 and 4.8, it is sufficient to exhibit a (2 − ǫ)approximate algorithm, for some constant ǫ > 0, for VC-PM. This would
give us the desired contradiction. We show that A itself is such an algorithm
and obtains a (2 − ǫ)-approximate solution with ǫ = 2/(c + 1).
Let G = (V, E) be a graph on 2n vertices with a perfect matching and
minimum vertex cover of size n + αn, where 1/n ≤ α ≤ 1. Use algorithm A
on G to obtain a vertex cover of size at most n + cαn. The quality of this
solution is (n + cαn)/(n + αn) = (1 + cα)/(1 + α). We distinguish two cases:
(1) α < 1/c and (2) α ≥ 1/c. We claim that in either case the approximation
factor is 2 − 2/(c + 1) = 2c/(c + 1). To see this, first consider the case
when α < 1/c. It is straightforward to show that (1+cα)/(1+α) < 2c/(c+1)
if and only if α < 1/c. When α ≥ 1/c, note that A returns the entire vertex
set of G as solution. The approximation factor in this case is 2/(1+α) which
can be easily seen to be at most 2c/(c + 1). This completes the proof of the
theorem.
Since there is an approximation-preserving reduction (Lemma 4.5) from
Above Guarantee Vertex Cover to Min 2-Cnf Sat Del, a constantfactor approximation algorithm for the latter implies the existence of a
constant-factor approximation algorithm for the former. Thus we have,
Corollary 4.1. If UGC is true then Min 2-Cnf Sat Del does not admit
a c-factor approximation algorithm, for any constant c > 1.
To the best of our knowledge, the only other hardness result known for
Min 2-Cnf Sat Del is a 2.88-approximation hardness assuming P 6= NP
due to Chlebik and Chlebikova [35].
62
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
Dinur and Safra [45] have shown that unless P = NP, Vertex Cover
cannot be approximated to within 1.3606 even on graphs with a perfect
matching. Using this, we obtain:
Corollary 4.2. Under the hypothesis P 6= NP, Above Guarantee Vertex Cover in graphs with a perfect matching cannot be approximated to
within 1.7212.
Proof. Let A be a d-approximation algorithm for computing τ (G) in graphs
with a perfect matching. Let G be an n-vertex graph with a perfect matching. Using A, one can obtain a vertex cover of size at most n/2 + dτ (G).
An optimum vertex cover of G has size n/2 + τ (G). By the NP-hardness
of approximating Vertex Cover [45], we must have (n + 2dτ (G))/(n +
2τ (G)) ≥ 1.3606. Simplifying this yields n/τ (G) ≤ 2(d − 1.3606)/0.3606.
Note that n/τ (G) ≥ 2 and so d ≥ 1.7212.
4.4
The Kőnig Vertex Deletion Problem
Recall that the Kőnig Vertex Deletion problem is, given a graph and
a nonnegative integer parameter k, to decide whether there exist at most k
vertices whose deletion makes the resulting graph Kőnig. We first investigate the parameterized complexity of this problem and then describe an
approximation algorithm for its optimization version.
4.4.1
Parameterized Complexity
We first consider the case when the input graph has a perfect matching.
Graphs with a Perfect Matching
For graphs with a perfect matching it turns out that Kőnig Vertex Deletion and AGVC are fixed-parameter equivalent.
Theorem 4.7. Let G be an n-vertex graph with a perfect matching. Then G
has a vertex cover of size at most n/2+ k if and only if G has a Kőnig vertex
deletion set of size at most 2k.
Proof. (⇒) Let P be a perfect matching of G and C a vertex cover of G of
size at most n/2 + k. Consider the subset M ⊆ P of matching edges both
of whose endpoints are in C. Clearly V [M ] is a Kőnig vertex deletion set
of G of size at most 2k.
(⇐) Conversely let K be a Kőnig vertex deletion set of G of size r ≤ 2k.
Then G \ K is a Kőnig graph on n − r vertices and hence has a vertex
cover C ′ of size at most (n − r)/2. Clearly C = C ′ ∪ K is a vertex cover
of G of size |C ′ | + |K| ≤ (n − r)/2 + r = (n + r)/2 ≤ n/2 + k.
4.4. The Kőnig Vertex Deletion Problem
63
The following corollary follows from Theorems 4.1 and 4.7 and the fact
that Vertex Cover is NP-complete.
Corollary 4.3. The Kőnig Vertex Deletion problem is NP-complete.
By Theorem 4.3, AGVC is fixed-parameter tractable and therefore we
have:
Corollary 4.4. The Kőnig Vertex Deletion problem, parameterized
by the solution size, is fixed-parameter tractable on graphs with a perfect
matching.
The next result relates the size of a minimum vertex cover with that of
a minimum Kőnig vertex deletion set for graphs with a perfect matching.
Corollary 4.5. Let G be an n-vertex graph with a perfect matching P . Then
β(G) = n/2 + k if and only if κ(G) = 2k. Moreover if κ(G) = 2k, then there
exists an edge subset M ⊆ P of size k such that V [M ] is a minimum Kőnig
vertex deletion set of G.
If we let τ (G) = β(G) − µ(G), then the above corollary states: κ(G) =
2τ (G).
Graphs Without a Perfect Matching
For graphs without a perfect matching we do not know of a reduction from
Kőnig Vertex Deletion to AGVC and neither does the general case seem
reducible to the case where the graph has a perfect matching. However we
show that the general problem is fixed-parameter tractable using some new
structural results between maximum matchings and vertex covers.
To begin with, we derive a weaker version of Theorem 4.7 which relates
the size of a vertex cover with that of a Kőnig vertex deletion set for graphs
without a perfect matching.
Theorem 4.8. Let G be a graph without a perfect matching. If G has a
vertex cover of size µ(G) + k then G has a Kőnig vertex deletion set of size
at most 2k. Moreover, τ (G) ≤ κ(G) ≤ 2τ (G), where τ (G) = β(G) − µ(G).
Proof. Let M be a maximum matching of G and let C be a vertex cover
of G of size µ(G) + k. Define I = V \ V [M ], CI = C ∩ I and M ′ to be the
subset of M both of whose endpoints are in C. Clearly V [M ′ ]∪CI is a Kőnig
vertex deletion set of G of size at most 2k. This shows that κ(G) ≤ 2τ (G).
To prove that τ (G) ≤ κ(G), suppose that there exists S ⊆ V , |S| < τ (G),
such that G \ S is Kőnig. Then the following easily verifiable inequalities:
µ(G \ S) ≤ µ(G)
β(G \ S) ≥ β(G) − |S| = µ(G) + τ (G) − |S|
imply that β(G \ S) > µ(G \ S), a contradiction.
64
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
K
K′
A1
V1′
M
V2′
A2
V1
V2
Figure 4.2: The sets that appear in the proof of Theorem 4.9. The matching M
consists of the solid edges across V1 and V2 .
Suppose Y is a vertex cover in a graph G = (V, E). Consider a maximum
matching M between Y and V \Y . If M saturates every vertex of Y then the
graph is Kőnig. If not, then Y \ V (M ), the set of vertices of Y unsaturated
by M , is a Kőnig vertex deletion set by Lemma 4.2. What we prove in this
section is that if Y is a minimum vertex cover, then Y \ V (M ) is a minimum
Kőnig vertex deletion set. Our first observation is that any minimum Kőnig
vertex deletion set is contained in some minimum vertex cover.
Theorem 4.9. Let G be a graph with a minimum Kőnig vertex deletion
set K. Let V (G \ K) = V1 ⊎ V2 where V2 is independent and there is a
matching M from V1 to V2 saturating V1 . Then V1 ∪ K is a minimum vertex
cover for G.
Proof. Suppose S is a vertex cover of G such that |S| < |V1 | + |K|. We will
show that there exists a Kőnig vertex deletion set of size smaller than |K|,
contradicting our hypothesis. Define V1′ = V1 ∩ S, V2′ = V2 ∩ S and K ′ =
K ∩ S. Let A1 be the vertices of V1′ whose partner in M is in V2′ and let A2
be the vertices of V1′ whose partner in M is not in V2′ . See Figure 4.2. We
claim that A1 ∪ K ′ is a Kőnig vertex deletion set of G and |A1 ∪ K ′ | < |K|,
which will produce the required contradiction and prove the theorem. This
claim is proved using the following three claims:
Claim 1. |A1 ∪ K ′ | < |K|.
Claim 2. A2 ∪ V2′ is a vertex cover in G \ (A1 ∪ K ′ ).
Claim 3. There exists a matching between A2 ∪ V2′ and V \ (V1′ ∪ K ′ ∪ V2′ )
saturating every vertex of A2 ∪ V2′ .
Proof of Claim 1. Clearly |S| = |V1′ |+|V2′ |+|K ′ |. Note that S intersects |A1 |
of the edges of M in both end points and |M | − |A1 | edges of M in one end
point (in either V1′ or V2′ ). Furthermore V2′ has |V2′ \ V (M )| vertices of S
that do not intersect any edge of M . Hence
|M | + |A1 | + |V2′ \ V (M )| = |V1′ | + |V2′ |.
4.4. The Kőnig Vertex Deletion Problem
65
That is,
|V1 | + |A1 | + |V2′ \ V (M )| = |V1′ | + |V2′ |,
as |M | = |V1 |. Hence |S| < |V1 | + |K| implies that
|A1 | + |V2′ \ V (M )| + |K ′ | < |K|
which implies that |A1 | + |K ′ | < |K| proving the claim.
Proof of Claim 2. Since S = A1 ∪ A2 ∪ V2′ ∪ K ′ is a vertex cover of G,
clearly A2 ∪ V2′ covers all edges in G \ (A1 ∪ K ′ ).
Proof of Claim 3. Since the partner of a vertex in A2 in M is in V \ (V1′ ∪
K ′ ∪ V2′ ), we can use the edges of M to saturate vertices in A2 . To complete
the proof, we show that in the bipartite graph
B = (V2′ , (V1 \ V1′ ) ∪ (K \ K ′ ))
there is a matching saturating V2′ . To see this, note that any subset D ⊆ V2′
has at least |D| neighbors in (V1 \ V1′ ) ∪ (K \ K ′ ). For otherwise, let D ′ be
the set of neighbors of D in (V1 \ V1′ ) ∪ (K \ K ′ ) where we assume |D| > |D ′ |.
Then (S \ D) ∪ D ′ is a vertex cover of G of size strictly less than |S|,
contradicting the fact that S is a minimum vertex cover. To see that (S \
D) ∪ D ′ is indeed a vertex cover of G, note that S \ V2′ covers all edges of G
except those in the graph B and all these edges are covered by (V2′ \ D) ∪ D ′ .
Hence by Hall’s theorem, there exists a matching saturating all vertices of
V2′ in the bipartite graph B, proving the claim.
This completes the proof of the theorem.
Theorem 4.9 has interesting consequences.
Corollary 4.6. If K1 and K2 are minimum Kőnig vertex deletion sets of G,
then µ(G \ K1 ) = µ(G \ K2 ).
Proof. Since K1 and K2 are minimum Kőnig vertex deletion sets of G, β(G\
K1 ) = µ(G \ K1 ) and β(G \ K2 ) = µ(G \ K2 ). By Theorem 4.9, β(G \ K1 ) +
|K1 | = β(G) and β(G \ K2 ) + |K2 | = β(G). Since |K1 | = |K2 |, it follows
that β(G \ K1 ) = β(G \ K2 ) and hence µ(G \ K1 ) = µ(G \ K2 ).
From Theorem 4.9 and Lemma 4.4, we get
Corollary 4.7. Given a graph G = (V, E) and a minimum Kőnig vertex
deletion set for G, one can construct a minimum vertex cover for G in
polynomial time.
Our goal now is to prove the “converse” of Corollary 4.7. In particular,
we would like to construct a minimum Kőnig vertex deletion set from a
minimum vertex cover. Our first step is to show that if we know that a given
minimum vertex cover contains a minimum Kőnig vertex deletion set then we
66
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
can find the Kőnig vertex deletion set in polynomial time. Recall that given
a graph G = (V, E) and A, B ⊆ V such that A ∩ B = ∅, we use µ(A, B)
to denote a maximum matching in the bipartite graph comprising of the
vertices in A ∪ B and the edges in {{u, v} ∈ E : u ∈ A, v ∈ B}. We denote
this graph by (A, B).
Lemma 4.9. Let K be a minimum Kőnig vertex deletion set and Y a minimum vertex cover of a graph G = (V, E) such that K ⊆ Y . Then µ(G\K) =
µ(Y, V \ Y ) and |K| = |Y | − µ(Y, V \ Y ).
Proof. If G is Kőnig then the theorem clearly holds. Therefore assume
that K 6= ∅. Note that Y \ K is a minimum vertex cover of the Kőnig
graph G \ K. Thus µ(G \ K) = µ(Y \ K, V \ Y ). We claim that µ(Y \ K, V \
Y ) = µ(Y, V \ Y ). For if not, we must have µ(Y \ K, V \ Y ) < µ(Y, V \ Y ).
Then let M be a maximum matching in the bipartite graph (Y, V \ Y )
and K ′ ⊆ Y be the set of vertices unsaturated by M . Note that K ′ 6=
∅ is a Kőnig vertex deletion set for G. Since µ(Y, V \ Y ) = |Y | − |K ′ |
and µ(Y \ K, V \ Y ) = |Y | − |K| we have |K ′ | < |K|, a contradiction, since
by hypothesis K is a smallest Kőnig vertex deletion set for G. Therefore we
must have µ(G \ K) = µ(Y, V \ Y ) and |K| = |Y | − µ(Y, V \ Y ).
The next lemma says that µ(Y, V \Y ) is the same for all minimum vertex
covers Y of a graph G. Together with Lemma 4.9, this implies that if K is
a minimum Kőnig vertex deletion set and Y is a minimum vertex cover of
a graph G = (V, E), then µ(G \ K) = µ(Y, V \ Y ).
Lemma 4.10. For any two minimum vertex covers Y1 and Y2 of G, µ(Y1 , V \
Y1 ) = µ(Y2 , V \ Y2 ).
Proof. Suppose without loss of generality that µ(Y1 , V \ Y1 ) > µ(Y2 , V \ Y2 ).
Let M1 be a maximum matching in the bipartite graph (Y1 , V \ Y1 ). To
arrive at a contradiction, we study how Y2 intersects the sets Y1 and V \ Y1
with respect to the matching M1 . To this end, we define the following sets
(see Figure 4.3):
• A = Y2 ∩ Y1 ∩ V (M1 ).
• B = Y2 ∩ (V \ Y1 ) ∩ V (M1 ).
• A1 is the set of vertices in A whose partners in M1 are also in Y2 .
• A2 is the set of vertices in A whose partners in M1 are not in Y2 .
We first show that
Claim. In the bipartite graph (Y2 , V \ Y2 ) there is a matching saturating
each vertex in A2 ∪ B.
4.4. The Kőnig Vertex Deletion Problem
67
A1
M1
A2
P
B
Y1
V \ Y1
Figure 4.3: The sets that appear in the proof of Lemma 4.10. The solid edges
across Y1 and V \ Y1 constitute the matching M1 .
It will follow from the claim that µ(Y2 , V \ Y2 ) ≥ |A2 | + |B|. However,
note that Y2 intersects every edge of M1 at least once (as Y2 is a vertex
cover). More specifically, Y2 intersects |A1 | edges of M1 twice and |M1 |−|A1 |
edges once (either in Y1 or in V \ Y1 ). Hence, |A| + |B| = |M1 | + |A1 | and
so |A2 | + |B| = |M1 | and so µ(Y2 , V \ Y2 ) ≥ |A2 | + |B| = |M1 | a contradiction
to our assumption at the beginning of the proof. Thus it suffices to prove
the claim.
Proof of Claim. Let P denote the partners of the vertices of A2 in M1 .
Since P ⊆ V \ Y2 , we use the edges of M1 to saturate vertices of A2 . Hence
it is enough to show that the bipartite graph B = (B, (V \ Y2 ) \ P ) contains
a matching saturating the vertices in B. Suppose not. By Hall’s Theorem
there exists a set D ⊆ B such that |NB (D)| < |D|. We claim that the
set Y2′ := Y2 \ D + NB (D) is a vertex cover of G. To see this, note that
the vertices in Y2 \ D cover all the edges of G except those in the bipartite
graph (D, Y1 ∩ (V \ Y2 )) and these are covered by NB (D). Therefore Y2′ is
a vertex cover of size strictly smaller than Y2 , a contradiction. This proves
that there exists a matching in (Y2 , V \ Y2 ) saturating each vertex in A2 ∪ B.
This completes the proof of the lemma.
The next theorem shows how we can obtain a minimum Kőnig vertex
deletion set from a minimum vertex cover in polynomial time.
Theorem 4.10. Given a graph G = (V, E), let Y be any minimum vertex
cover of G and M a maximum matching in the bipartite graph (Y, V \ Y ).
Then K := Y \ V (M ) is a minimum Kőnig vertex deletion set of G.
Proof. Clearly K is a Kőnig vertex deletion set. Let K1 be a minimum
Kőnig vertex deletion set of G. By Theorem 4.9, there exists a minimum
vertex cover Y1 such that K1 ⊆ Y1 and
|K1 | =
=
=
=
|Y1 | − µ(Y1 , V \ Y1 ) (By Lemma 4.9.)
|Y | − µ(Y1 , V \ Y1 )
|Y | − µ(Y, V \ Y )
(By Lemma 4.10.)
|K|
68
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
The second equality holds since Y1 and Y are minimum vertex covers. This
proves that K is a minimum Kőnig vertex deletion set.
Corollary 4.8. Given a graph G = (V, E) and a minimum vertex cover
for G, one can construct a minimum Kőnig vertex deletion set for G in
polynomial time.
Note that although both these problems—Vertex Cover and Kőnig
Vertex Deletion—are NP-complete, we know of very few pairs of such
parameters where we can obtain one from the other in polynomial time on
the same graph (e.g. edge dominating set and minimum maximal matching,
see [66]). In fact, there are parameter pairs such as dominating set and
vertex cover where such a polynomial-time transformation is not possible
unless P = NP. This follows since in bipartite graphs, for instance, a minimum vertex cover is computable in polynomial time whereas computing a
minimum dominating set is NP-complete.
We are now ready to prove that the Kőnig Vertex Deletion problem
is fixed-parameter tractable in general graphs.
Theorem 4.11. Given a graph G = (V, E) and an integer parameter k,
the problem of whether G has a subset of at most k vertices whose deletion
makes the resulting graph Kőnig can be decided in time O(15k · k2 · |E|3 ).
Hence the Kőnig Vertex Deletion problem is fixed-parameter tractable
when parameterized by the solution size.
Proof. Use the FPT-algorithm from Theorem 4.3 to test whether G has a
vertex cover of size at most µ(G) + k. If not, by Theorem 4.8, we know
that the size of a minimum Kőnig vertex deletion set is strictly more than k.
Therefore return no. If yes, then find the size of a minimum vertex cover by
applying Theorem 4.3 with every integer value between 0 and k for the excess
above µ(G). Note that for yes-instances of the Above Guarantee Vertex Cover problem, the FPT-algorithm actually outputs a vertex cover
of size µ(G) + k. We therefore obtain a minimum vertex cover of G. Use
Theorem 4.10 to get a minimum Kőnig vertex deletion set in polynomial
time and depending on its size answer the question. It is easy to see that
all this can be done in time O(15k · k2 · |E|3 ).
Note that computing a maximum independent set (or equivalently a minimum vertex cover) in an n-vertex graph can be done in time O ∗ (20.288n ) [62].
By Corollary 4.8, one can compute a minimum Kőnig vertex deletion set in
the same exponential time.
Corollary 4.9. Given a graph G = (V, E) on n vertices one can find a
minimum Kőnig vertex deletion set in time O ∗ (20.288n ) = O ∗ (1.221n ).
Suppose we wanted to compute a minimum Kőnig vertex deletion set
on graphs of treewidth at most w. A dynamic programming approach as
4.5. The Induced Kőnig Subgraph Problem
69
for Dominating Set or Independent Set is not obvious. However since
one can obtain a minimum vertex cover on graphs with treewidth at most w
in time O ∗ (2w ) [107], by Corollary 4.8, one can obtain a minimum Kőnig
deletion set within this time.
Corollary 4.10. If a tree-decomposition for G of width w is given, one can
find a minimum Kőnig vertex deletion set in time O ∗ (2w ).
4.4.2
Approximability
In Theorem 4.8 we established that for any graph G (whether it has a perfect
matching or not), τ (G) ≤ κ(G) ≤ 2τ (G), where τ (G) = β(G) − µ(G) is the
excess vertex cover beyond the size of a maximum matching. Therefore
a good approximation of the above guarantee parameter τ yields a good
approximation for κ and vice versa.
In the algorithm outlined in Figure 4.1, note that V (S)∩V (G) is actually
a Kőnig vertex deletion set of G. Since |V (S)| ≤ O(log n log log n) (β − µ),
we have, by Theorem 4.4
Theorem 4.12. Given a graph G on n vertices, there exists an algorithm
that approximates the Kőnig vertex deletion set of G to within a factor of
O(log n log log n).
On graphs with a perfect matching, the Above Guarantee Vertex
Cover and Kőnig Vertex Deletion problems are equivalent and hence
Theorems 4.6 and 4.7 imply
Corollary 4.11. If UGC is true then Kőnig Vertex Deletion does not
admit a constant-factor approximation algorithm.
Since Kőnig Vertex Deletion and Above Guarantee Vertex
Cover are equivalent in terms of approximability in graphs with a perfect
matching (Theorem 4.7), Corollary 4.2 implies
Corollary 4.12. Under the hypothesis P 6= NP, Kőnig Vertex Deletion
in graphs with a perfect matching cannot be approximated to within 1.7212.
4.5
The Induced Kőnig Subgraph Problem
In this section we deal with the parameterized complexity and approximability of the vertex and edge versions of the Induced Kőnig Subgraph
problem.
70
4.5.1
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
Vertex Induced Kőnig Subgraph
The NP-completeness of this problem follows from that of Kőnig Vertex Deletion but it has a different parameterized complexity. We show
that Vertex Induced Kőnig Subgraph is W[1]-hard and is as hard to
approximate as the Independent Set problem.
Theorem 4.13. Vertex Induced Kőnig Subgraph is W[1]-hard with
respect to the number of vertices in the induced subgraph as parameter.
Proof. We give a parameter-preserving reduction from Independent Set
to Vertex Induced Kőnig Subgraph. Given an instance (G, k) of Independent Set, construct a graph H as follows. The vertex set of H consists
of two copies of V (G) namely, V1 = {u1 : u ∈ V (G)} and V2 = {u2 : u ∈
V (G)}. For all u ∈ V (G), (u1 , u2 ) ∈ E(H). If (u, v) ∈ E(G), add the edges
(u1 , v1 ), (u2 , v2 ), (u1 , v2 ) and (v1 , u2 ) in E(H). H has no more edges.
We claim that G has an independent set of size k if and only if H
has a Kőnig subgraph of size 2k. Let I be an independent set of size k
in G. Let K = {u1 , u2 ∈ V (H) : u ∈ I}. Clearly H[K] is an induced
matching on 2k vertices and is bipartite and hence Kőnig. Conversely, let K
be a Kőnig subgraph of H on 2k vertices. By Lemma 4.2, every Kőnig
graph on n vertices has an independent set of size at least n/2. Therefore
let I ′ be an independent set of K of size at least k. Define I = {u ∈
V (G) : either u1 or u2 ∈ I ′ }. It is clear that the vertices of I ′ correspond
to distinct vertices of G and hence |I| ≥ k. It is also easy to see that the
vertices in I actually form an independent set in G.
Since the Independent Set problem can have no approximation algorithms with factor O(n1−ǫ ), for any ǫ > 0, unless P = NP [80, 127], we
have:
Corollary 4.13. The Vertex Induced Kőnig Subgraph problem cannot be approximated in polynomial time to within a factor of O(n1−ǫ ), for
any ǫ > 0, unless P = NP.
In the reduction above, |V (H)| = 2|V (G)| and (G, k) is a yes-instance
of Independent Set if and only if (H, 2k) is a yes-instance of Vertex
Induced Kőnig Subgraph. Thus this reduction can also be viewed as a
reduction from Vertex Cover to Kőnig Vertex Deletion giving yet
another proof of Corollary 4.3.
4.5.2
Edge Induced Kőnig Subgraph
In this section we study the Edge Induced Kőnig Subgraph problem.
We begin by showing that it is NP-complete.
4.5. The Induced Kőnig Subgraph Problem
71
NP-Completeness
Since both Kőnig Edge Deletion and Edge Induced Kőnig Subgraph
have the same complexity from the classical point of view, it is sufficient to
prove that one of them is NP-complete. We actually show that:
Theorem 4.14. Kőnig Edge Deletion is NP-complete.
Proof. We give a reduction from Min 2-Cnf Sat Del. Let Φ be a 2-Cnf
Sat formula with m clauses composed of the literals {x1 , x̄1 , . . . , xn , x̄n }.
Construct a graph GΦ = (V, E) as follows. The vertex set V consists of m+3
copies of xi , x̄i , for 1 ≤ i ≤ n:
xi , x̄i , xi1 , x̄i1 , . . . , xi,m+2 , x̄i,m+2 .
Add exactly those edges so that for each 1 ≤ i ≤ n, the vertex sets Li =
{xi , xi,1 , . . . , xi,m+2 } and Ri = {x̄i , x̄i,1 , . . . , x̄i,m+2 } form a complete bipartite graph with Li and Ri as the left and right partite sets, respectively.
Finally for each clause (yi ∨ yj ) of Φ add an edge (yi , yj ) (among the vertices {x1 , x̄1 , . . . , xn , x̄n }). Note that GΦ has a perfect matching and that
each clause of Φ corresponds to an edge of GΦ .
Claim 4.1. There exists an assignment satisfying all but k clauses of Φ if
and only if there exist at most k edges whose deletion makes GΦ Kőnig.
(⇒) Let α be an assignment to the variables of Φ that satisfies all but k
clauses. Each of these k clauses corresponds to a distinct edge in GΦ . Delete
these edges from GΦ . Then for each edge in the remaining graph, at least one
endpoint of the edge is assigned 1 by the assignment α. To prove that the
remaining graph is Kőnig, by Lemma 4.2, we must demonstrate a bipartition
of the vertex set into V1 ⊎ V2 (say) such that V2 is independent and there
exists a matching across the cut (V1 , V2 ) saturating V1 . If α(xi ) = 1 then
place the vertices xi , xi,1 , . . . xi,m+2 in V1 ; else place x̄i , x̄i,1 , . . . x̄i,m+2 in V1 .
The remaining vertices are placed in V2 . As Φ satisfies all remaining clauses,
V2 is independent. Note that if xi ∈ V1 then x̄i ∈ V2 and vice versa. Also
if xi,j ∈ V1 then x̄i,j ∈ V2 and vice versa. Hence there exists a matching
across the cut (V1 , V2 ) that saturates V1 .
(⇐) Conversely suppose that deleting a set S of k edges makes GΦ Kőnig.
We will assume that S is a minimal edge deletion set. Any minimal Kőnig
edge deletion set has size at most m, since deleting all the m “clause edges”
from GΦ results in a Kőnig graph. Therefore we may assume that k ≤ m.
Call the resulting graph G′Φ . Then the vertex set of G′Φ can be partitioned
into V1 and V2 such that V2 is independent and there exists a matching
across the cut (V1 , V2 ) that saturates V1 .
Claim 1. For each 1 ≤ i ≤ n, it is not the case that xi , x̄i ∈ V2 .
72
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
Suppose that for some 1 ≤ i ≤ n, xi , x̄i ∈ V2 . Then it must be that
C̄i = {x̄i,1 , . . . , x̄i,m+2 } * V2 since otherwise m + 2 edges between xi and the
vertices of C̄i must have been deleted from GΦ to obtain G′Φ , a contradiction.
If C̄i ⊆ V1 then Ci = {xi,1 , . . . , xi,m+2 } ⊆ V2 for there to be a matching
across the cut (V1 , V2 ) saturating all of C̄i . But then m + 2 edges between x̄i
and Ci must have been deleted from GΦ to obtain G′Φ , again a contradiction.
This argument shows that that there exist integers p, q ≥ 1 with p + q =
m + 2, such that V1 contains p vertices of C̄i and V2 contains the remaining q
vertices of C̄i . In order for there to be a matching across (V1 , V2 ) saturating
all p vertices of C̄i in V1 there must be at least p vertices of Ci in V2 . Since
the vertices of Ci and C̄i form a complete bipartite graph we end up deleting
at least pq + 1 ≥ m + 2 edges of GΦ , a contradiction yet again. This proves
Claim 1.
Claim 2. For each 1 ≤ i ≤ n, it is not the case that xi , x̄i ∈ V1 .
Suppose that there exists i, 1 ≤ i ≤ n, such that xi , x̄i ∈ V1 . Let M be
a matching across (V1 , V2 ) that saturates the vertices of V1 . We distinguish
three cases.
Case 1. C̄i = {x̄i,1 , . . . , x̄i,m+2 } ⊆ V1 .
This implies that Ci = {xi,1 , . . . , xi,m+2 } ⊆ V2 as otherwise no matching
across (V1 , V2 ) would saturate all of C̄i . Let xa1 and xb1 be the partners
of xi and x̄i , respectively, relative to the matching M . By Claim 1, xa1
and xb1 represent different variables, that is, they are not the negations
of one another. This implies that x̄a1 and x̄b1 are in V1 . Consider the
pair xa1 , x̄a1 . We will show that Ca1 = {xa1 ,1 , . . . , xa1 ,m+2 } ⊆ V2 and C̄a1 =
{x̄a1 ,1 , . . . , x̄a1 ,m+2 } ⊆ V1 . For if not, suppose that 1 ≤ q ≤ m + 1 vertices
of C̄a1 are in V2 while the remaining p ≥ 1 vertices of C̄a1 are in V1 . In
order for the vertices of C̄a1 to have partners with respect to M at least p
vertices of Ca1 must be in V2 . This implies that at least pq edges have been
deleted from GΦ to obtain G′Φ . Since p + q = m + 2, we have pq ≥ m + 1, a
contradiction. The upshot is that the partners of x̄a1 and x̄b1 relative to M
are vertices from the set {x1 , x̄1 , . . . , xn , x̄n }. Let the partners of x̄a1 and x̄b1
relative to M be xa2 and xb2 respectively. Again by Claim 1, xa2 and xb2
represent distinct variables and hence x̄a2 and x̄b2 are in V1 . Repeating this
argument we obtain a sequence of vertices of the form:
xi
..
.
xa1
..
.
x̄a1
..
.
xa2
..
.
x̄a2
..
.
...
x̄i
xb1
x̄b1
xb2
x̄b2
...
V1
V2
V1
V2
V1
Since there are only 2n vertices such a chain must end at V2 with both
endpoints being the negation of one another. This contradicts Claim 1 and
shows that this situation does not arise.
4.5. The Induced Kőnig Subgraph Problem
73
Case 2. C̄i = {x̄i,1 , . . . , x̄i,m+2 } ⊆ V2 and Ci = {xi,1 , . . . , xi,m+2 } ⊆ V1 .
This is the symmetric version of Case 1 and can be handled similarly.
Case 3. For some integers p, q ≥ 1 and p + q = m + 2, p vertices of C̄i lie
in V1 and the remaining q vertices lie in V2 .
By symmetry, p vertices of Ci must lie in V2 . This implies that at
least pq ≥ m + 1 edges have been deleted from GΦ to obtain G′Φ , a contradiction. This proves Claim 2.
Since S was assumed to be a minimal Kőnig edge deletion set, for each
vertex yi , all copies yi,1 , . . . , yi,m+2 of it must be placed in the same partition as yi itself and hence all edges of GΦ that are part of the n copies
of Km+3,m+3 lie across the cut (V1 , V2 ). It is easy to see that any other partitioning of the copies would result in more edges being deleted unnecessarily.
Therefore the edges that were deleted from GΦ were those that corresponded
to the clauses of Φ. If a vertex yi is in V1 assign the corresponding literal
the value 1; else assign the literal the value 0. Note that this assignment is
consistent as all copies of a vertex are in the same partition as the vertex
itself and for no vertex do we have that xi , x̄i ∈ V1 or xi , x̄i ∈ V2 . This
assignment satisfies all but the k clauses that correspond to the edges that
were deleted.
Since the above reduction is cost-preserving, an approximation lowerbound for Min 2-Cnf Sat Del is a lower-bound for Kőnig Edge Deletion too. Therefore by Corollary 4.1, we obtain
Corollary 4.14. If UGC is true then Kőnig Edge Deletion does not
have a c-approximation algorithm, for any constant c > 1.
Chlebik and Chlebikova [35] have shown
that it is NP-hard to approxi√
mate Min 2-Cnf Sat Del to within 8 5 − 15 ≈ 2.88. This gives us
Corollary 4.15. It is NP-hard to approximate Kőnig Edge Deletion to
within 2.88.
Approximation Results
For Edge Induced Kőnig Subgraph, an easy 2-approximation algorithm
is as follows: Find a cut of size m/2 and delete all other edges; the resulting graph is bipartite and hence Kőnig. In this subsection, we give a
4/3-approximation algorithm for graphs with a perfect matching and a 5/3approximation algorithm for general graphs based on the following combinatorial results.
Theorem 4.15. Let G = (V, E) be a graph with a maximum matching M
and let GM = (VM , EM ) be the graph induced on the vertices V (M ) of M .
Then G has an edge-induced Kőnig subgraph of size at least
3|EM | |E − EM | |M |
+
+
.
4
2
4
74
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
Proof. Randomly partition the vertex set of G into V1 ⊎ V2 as follows. For
each edge ei ∈ M , select an endpoint of ei with probability 1/2 and place it
in V1 . Define V2 = V − V1 . Note that the edges in M always lie across the
cut (V1 , V2 ). An edge of EM − M is in G[V2 ] with probability 1/4; an edge
in E − EM lies in G[V2 ] with probability 1/2. For each edge e ∈ E, define Xe
to be the indicator random variable
that takes the value 1 if e ∈ G[V2 ] and 0
P
otherwise. Also define X = e∈E Xe . Then
E[X] =
X
E[Xe ] =
e∈E
|EM − M | |E − EM |
+
.
4
2
Deleting the edges in G[V2 ] results in a Kőnig graph with
3|EM | |E − EM | |M |
+
+
4
2
4
edges in expectation. This algorithm can be easily derandomized by the
method of conditional probabilities (see, for instance [104]). This completes
the proof.
If G = (V, E) has a perfect matching M then EM = E and |M | = |V |/2
and we have
Corollary 4.16. Let G = (V, E) be a graph on n vertices and m edges with
a perfect matching. Then G has a subgraph with at least 3m/4 + n/8 edges
that is Kőnig. This subgraph can be found in time O(mn).
Theorem 4.16. Let G = (V, E) be an undirected graph on n vertices and m
edges. Then G has an edge-induced Kőnig subgraph of size at least 3m/5.
Proof. Let M be a maximum matching of G and let G[VM ] = (VM , EM ) be
the subgraph induced by the vertices V (M ) of M . Let η(G) denote the size
of the maximum edge induced Kőnig subgraph of G. By Theorem 4.15,
η(G) ≥
|EM | + |M | |E|
+
.
4
2
Observe that by deleting all the edges in G[VM ] we obtain a Kőnig subgraph
of G. In fact, this is a bipartite graph with bipartition VM and V − VM .
Therefore if |E − EM | ≥ 3m/5, the statement of the theorem clearly holds.
Otherwise, |EM | ≥ 2m/5 and by Theorem 4.15, we obtain η(G) ≥ |M |/4 +
3m/5. This completes the proof.
The following theorem follows from Corollary 4.16 and Theorem 4.16
and the fact that the optimum Kőnig subgraph has at most m edges.
Theorem 4.17. The optimization version of Edge Induced Kőnig Subgraph is approximable to within a factor of 5/3 for general graphs. This
factor can be improved to 4/3 when restricted to graphs with a perfect matching.
4.5. The Induced Kőnig Subgraph Problem
75
FPT Algorithms
Note that Theorem 4.16 actually shows that Edge Induced Kőnig Subgraph is fixed-parameter tractable. To see this, suppose that (G, k) is an instance of the problem; we are to decide whether G has an edge induced Kőnig
subgraph with at least k edges. Note that if the parameter k ≤ 3m/5 then we
answer yes and use the approximation algorithm described in the previous
subsection to obtain an edge induced Kőnig subgraph with at least k edges.
If k > 3m/5 then we simply use a trivial O∗ (2m ) brute-force algorithm to
decide the question. This FPT algorithm has time complexity O ∗ (25k/3 ).
In this subsection, we give an O ∗ (2k ) FPT algorithm for Edge Induced
Kőnig Subgraph on connected graphs by using an exact algorithm for the
optimization version of the problem. To this end, we describe an O∗ (2n )
algorithm for this problem using a simple structural result characterizing
minimal Kőnig edge deletion sets of a graph.
Theorem 4.18. Let G = (V, E) be a graph. If E ′ is a minimal Kőnig edge
deletion set of G then there exists V ′ ⊆ V such that E(G[V ′ ]) = E ′ , that is,
the edge set of the subgraph induced by V ′ is precisely E ′ .
Proof. Let E ′ be a minimal Kőnig edge deletion set of G. Then G′ =
(V, E − E ′ ) is Kőnig. Then the vertex set of G′ can be partitioned into
V1 and V2 such that V2 is a maximal independent set and there exists a
matching saturating V1 that lies across the cut (V1 , V2 ). Let V ′ = V2 . Since
E ′ is minimal, it is clear that E(G[V2 ]) = E ′ . This completes the proof.
Our exact algorithm for the optimization version of Edge Induced
Kőnig Subgraph simply enumerates all possible subsets V ′ ⊆ V , deletes
all edges E ′ in G[V ′ ] and checks whether G − E ′ is Kőnig. The algorithm
returns an edge set E ′ = E(G[V ′ ]) of smallest size such that G−E ′ is Kőnig.
Theorem 4.19. Given an n-vertex graph G = (V, E), the optimization
version of the Kőnig Edge Deletion (and hence the optimization version
of Edge Induced Kőnig Subgraph) can be solved in time O ∗ (2n ) and
space polynomial in n.
Theorem 4.20. Edge Induced Kőnig Subgraph can be solved in O ∗ (2k )
time in connected undirected graphs.
Proof. Let (G, k) be an instance of Edge Induced Kőnig Subgraph
where G is a graph with m edges and n vertices. A connected graph has a
spanning tree which, being bipartite, is Kőnig. Since a tree has n − 1 edges,
if k ≤ n−1 we answer yes; else n ≤ k +1 and we use Theorem 4.19 to obtain
an O ∗ (2k ) time algorithm for Edge Induced Kőnig Subgraph.
76
Chapter 4. Kőnig Graphs and Above-Guarantee Vertex Cover
Problem
Parameterized
Complexity
Approximability
KVD/
AGVC
FPT.
KED
Open.
VKS
EKS
W[1]-hard.
FPT.
O(log n log log n) approx. algorithm; NP-hard to approximate to within 1.7212; no constant-factor approx. algorithm assuming UGC.
NP-hard to approximate to within 2.88; no constantfactor approx. algorithm assuming UGC.
no factor-O(n1−ǫ ) approx. algorithm.
5/3-approx. algorithm for general graphs; 4/3approx. algorithm for graphs with a perfect matching
Table 4.1: List of problems dealt with in this chapter.
4.6
Conclusion and Open Problems
In this chapter we introduced and studied vertex and edge versions of the
Kőnig Subgraph problem from the points of view of parameterized complexity and approximation algorithms. Our results are summarized in Figure 4.1. We showed that Kőnig Vertex Deletion is in FPT whereas
Vertex Induced Kőnig Subgraph is W[1]-hard. The Edge Induced
Kőnig Subgraph problem is in FPT and we conjecture that Kőnig Edge
Deletion is W[1]-hard. Some obvious open problems are:
1. What is the parameterized complexity of the Kőnig Edge Deletion
problem?
2. Is there a better FPT-algorithm for Kőnig Vertex Deletion perhaps without making use of the algorithm for AGVC?
3. Are there better approximation algorithms for all these problems?
4. Theorem 4.15 shows that any graph G with a maximum matching M
has an edge-induced subgraph of size at least
3|EM | |E − EM | |M |
+
+
.
4
2
4
Is this lower bound tight? What is the parameterized complexity of
the above-guarantee question with respect to this lower bound?
Chapter 5
The Unique Coverage Problem
In this chapter we study the parameterized complexity of a problem known
as Unique Coverage, a variant of the classic Set Cover problem. This
problem is an analog of Longest Common Subsequence (see Chapter 1)
in that it admits several parameterizations and we show that all, except the
standard parameterization and a generalization of it, are unlikely to be fixedparameter tractable. During our study of the standard parameterized version of Unique Coverage we will have occasion to use several techniques.
We use results from extremal combinatorics to obtain the best-known kernel
for Unique Coverage and the well-known color-coding technique of Alon
et al. [6] to show that a weighted version of this problem is in FPT.
Our application of color-coding uses an interesting variation of s-perfect
hash families called (k, s)-hash families which were studied by Alon et al. [2]
in the context of a class of codes called parent identifying codes [14]. To
the best of our knowledge, this is the first application of (k, s)-hash families
outside the domain of coding theory. We prove the existence of such families
of size smaller than the best-known s-perfect hash families using the probabilistic method [5]. Explicit constructions of such families of size promised
by the probabilistic method is open.
5.1
Motivation and Known Results
The Unique Coverage problem was introduced by Demaine et al. [43] as
a natural maximization version of Set Cover and has applications in several areas including wireless networks and radio broadcasting. This problem
is defined as follows. Given a ground set U = {1, . . . , n}, a family of subsets F = {S1 , . . . , St } of U and a nonnegative integer k, does there exist a
subfamily F ′ ⊆ F such that at least k elements are covered uniquely by F ′ ?
An element is covered uniquely by F ′ if it appears in exactly one set of F ′ .
The optimization version requires us to maximize the number of uniquely
covered elements.
77
78
Chapter 5. The Unique Coverage Problem
We also consider the following weighted version of Unique Coverage,
called Budgeted Unique Coverage, defined as follows: Given a ground
set U = {1, . . . , n}, a profit pi for each element i ∈ U , a family F of subsets
of U , a cost ci for each set Si ∈ F, a budget B and a nonnegative integer k,
does there exist a subfamily F ′ ⊆ F with total cost at most B such that
the total profit of uniquely covered elements is at least k? The optimization
version asks for a subset F ′ of total cost at most B such that the total profit
of uniquely covered elements is maximized.
The original motivation for this problem is a real-world application arising in wireless networks [43]. Assume that we are given a map of the densities
of mobile clients along with a set of possible base stations, each with a specified building cost and a specified coverage region. The goal is to choose a
set of base stations, subject to a budget on the total building cost, in order
to maximize the density of served clients. The difficult aspect of this problem is the interference between base stations. A mobile client’s reception is
better when it is within the range of a few base stations. An ideal situation
is when every mobile client is within the range of exactly one base station.
This is the situation modelled by the Budgeted Unique Coverage problem. The Unique Coverage problem is closely related to a single “round”
of the Radio Broadcast problem [13]. For more about this relation, see
Demaine et al. [43].
One can also view the Unique Coverage problem as a generalization
of the Max Cut problem [43]. The input to the Max Cut problem consists
of a graph G = (V, E) and the goal is to find a cut (T, T ′ ), where ∅ =
6 T ⊂V
and T ′ = V \ T , that maximizes the number of edges with one endpoint
in T and the other endpoint in T ′ . Let U denote the set of edges of G
and for each
: e is incident to v}. Finally
S vertex v ∈ V define Sv = {e ∈ E
′ ) with at least k edges across
G
has
a
cut
(T,
T
let F = v∈V {Sv }. Then
S
it if and only if F ′ = v∈T {Sv } uniquely covers at least k elements of the
ground set.
Demaine et al. [43] considered the approximability of Unique Coverage. On the positive side, they give an Ω(1/ log n)-approximation for
Budgeted Unique Coverage. Moreover, if the ratio between the maximum cost of a set and the minimum profit of an element is bounded by b,
then there exists an Ω(1/ log b)-approximation algorithm for the weighted
version. They show that Unique Coverage is hard to approximate to
within a factor of O(1/ logc n) for some constant c depending on ǫ > 0,
ǫ
assuming NP * BPTIME(2n ) for some ǫ. They strengthen this inapproximability to 1/(ǫ log n) for some ǫ > 0 based on a hardness hypothesis for
Balanced Bipartite Independent Set.
Erlebach and van Leeuwen [51] study the approximability of geometric
versions of the Unique Coverage problem. Among the many versions that
they consider is Unique Coverage on Unit Disks, a variant in which
each set is a unit disk in R2 , for which they give a factor-18 approxima-
5.2. Results in this Chapter
79
tion algorithm. They also consider a variant called Unique Coverage on
Disks of Bounded Ply and design an asymptotic fully polynomial-time
approximation scheme (FPTASω ) for it.
5.2
Results in this Chapter
As with many problems in parameterized complexity, the Unique Coverage problem can be parameterized in a number of ways. We first consider
an extensive list of plausible parameterizations of the problem in Section 5.3
and discuss their parameterized complexity. Our results show that barring
the standard parameterized version (where the parameter is the number of
uniquely covered elements) and a generalization of it, the remaining parameterized problems are unlikely to be fixed-parameter tractable.
In Section 5.4 we consider the standard parameterized version of Unique
Coverage. We show that a special case of this version where any two sets in
the input family intersect in at most c elements is fixed-parameter tractable
by demonstrating a polynomial kernel of size kc+1 . This leads to a problem
kernel of size kk for the general case, proving that Unique Coverage is
fixed-parameter tractable. Then using results from extremal combinatorics
on strong systems of distinct representatives we obtain a 4k kernel. This is
essentially the best possible kernel for this problem since Dom et al. have
shown that Unique Coverage does not admit a polynomial kernel unless
the Polynomial Hierarchy collapses to the third level [46]. Note that the
size of an instance (U , F, k) of Unique Coverage is |U | + |F|. Therefore
when we say that there exists a kernel for Unique Coverage of size g(k),
what we mean is that there exists a kernelization algorithm that produces
an equivalent instance (U ′ , F ′ , k′ ) such that |U ′ | + |F ′ | ≤ g(k).
In Section 5.5 we consider the Budgeted Unique Coverage problem.
For this problem too, there are several variants. If the profits and costs are
allowed to be arbitrary positive rational numbers, then Budgeted Unique
Coverage, with parameters B and k, is not fixed-parameter tractable unless P = NP. If we restrict the costs and profits to be positive integers and
parameterize by B, then the problem is W[1]-hard. However if we parameterize with respect to both B and k then we show, using an application of
the color-coding technique, that the problem is fixed-parameter tractable.
In fact, we show that this remains true even when the costs are positive
integers and profits are rationals ≥ 1 or vice versa. While derandomizing
our algorithms, we use a variation of s-perfect hash families called (k, s)hash families and show the existence of such families of size smaller than
the best-known s-perfect hash families. We also modify the algorithms for
Budgeted Unique Coverage to two special cases: Unique Coverage
and Budgeted Max Cut.
80
Chapter 5. The Unique Coverage Problem
5.3
Unique Coverage: Which Parameterization?
First consider the following parameterized problem: given (U , F), find a
subfamily F ′ ⊆ F that covers all of U , each element being covered at most k
times and at least once, assuming k as parameter. This is a practical parameterization for mobile-computing applications. Unfortunately, this problem
is not fixed-parameter tractable unless P = NP as the case k = 1 reduces to
the NP-complete Exact Cover problem [64].
An alternate parameterization with a similar motivation of covering each
element a small number of times is as follows: given (U , F), find a subfamily F ′ ⊆ F of size at most |F| − k that covers all of U . Call this problem
All But k Coverage.
We show that this problem is W[1]-hard by a fixed-parameter reduction
from the W[1]-complete Red/Blue Nonblocker problem [47]:
A bipartite graph G = (R ⊎ B = V, E) with its vertex set
partitioned into a red set R and a blue set B, and a nonnegative integer k.
Parameter: The integer k.
Input:
Question:
Does there exist a set T of at least k red vertices such that
the vertices in R \ T dominate all vertices in B?
Theorem 5.1. All But k Coverage is W[1]-hard with respect to k as
parameter.
Proof. Given an instance (G = (R ⊎ B, E), k) of Red/Blue Nonblocker,
let (U , F, k′ ) be an instance of All But k Coverage defined as follows:
U := B, F := R, where each red vertex is interpreted as the set of blue
vertices it dominates in G, and k′ = k. Now it is easy to see that there exist
at least k red vertices such that the remaining red vertices dominate all blue
vertices if and only if there exists a subfamily of F size at most |F| − k that
covers all of U .
We also note that the following parameterized versions of Unique Coverage are unlikely to be fixed-parameter tractable. Given (U , F) and nonnegative integers k and t as parameters,
1. Does there exist a subfamily F ′ ⊆ F of size at most k that covers all
of U with each element being covered at most t times? This version
is not fixed-parameter tractable as the case t = 1 is W[1]-hard by a
reduction from Perfect Code [28] as shown below.
2. Does there exist a subfamily F ′ ⊆ F of size at most k that covers all
of U with each element being covered at most |F| − t times? This is
W[2]-hard as the case t = 0 reduces to the Set Cover problem which
is W[2]-complete.
5.4. Unique Coverage: The Standard Version
81
Call the version in Item 1 above with t = 1 the Disjoint Set Cover
problem which we now show to be W[1]-hard by a reduction from Perfect
Code [28].
Perfect Code
Input:
A graph G = (V, E) and an integer k.
Parameter: The integer k.
Question: Does G have a k-element perfect code?
A perfect code is a vertex subset V ′ ⊆ V such that for all u ∈ V we
have |N [u] ∩ V ′ | = 1, where N [u] is the closed neighborhood of vertex u,
that is, every vertex is dominated by exactly one vertex in V ′ . Given an instance (G, k) of Perfect Code construct an instance (U , F, k) of Disjoint
Set Cover by setting U := V (G) and F := {N [v] : v ∈ V }.
Lemma 5.1. The graph G has a k-element perfect code if and only if there
exists a subfamily F ′ ⊆ F of pairwise disjoint sets and of size at most k that
covers all of U .
Proof. Let {v1 , . . . , vk } ⊆ V be a k-element perfect code of G. Then clearly
S
k
i=1 N [vi ] covers all of U and the sets N [v1 ], . . . , N [vk ] are pairwise disjoint.
For if x ∈ N [vi ] ∩ N [vj ], i 6= j, then |N [x] ∩ {v1 , . . . , vk }| ≥ 2, which
contradicts the definition of a perfect code. Conversely if N [v1 ], . . . , N [vk ] is
a collection of pairwise disjoint sets that covers all of U then clearly v1 , . . . , vk
is a k-element perfect code for G.
Theorem 5.2. Disjoint Set Cover is W[1]-hard with respect to the number of sets in the solution as parameter.
Finally we consider a generalization of the standard parameterized version: Gen Unique Coverage: Given (U , F) and nonnegative integers k
and t, does there exist a subfamily of F that covers k elements at least once
and at most t times, where k is the parameter? Setting t = 1 gives us the
standard parameterized version of Unique Coverage. We note that Gen
Unique Coverage is fixed-parameter tractable as the kernel result for the
standard parameterized version works for this problem also. We elaborate
this further in Section 5.4.2.
5.4
Unique Coverage: The Standard Version
In this section we study the standard parameterized version of Unique
Coverage. Let (U = {1, . . . , n}, F = {S1 , . . . , Sm }, k) be an instance of
this problem, where k is the parameter. Apply the following rules on this
instance until no longer applicable.
82
Chapter 5. The Unique Coverage Problem
R1 If there exists Si ∈ F such that |Si | ≥ k, then the given instance is a
yes-instance.
R2 If there exists S1 , S2 ∈ F such that S1 = S2 , then delete S1 .
It is easy to see that these are indeed reduction rules for Unique Coverage.
For in the first case Si is a solution and in the second case, it is clear that
no solution need have both S1 and S2 . In the following we always assume
that the given instance of Unique Coverage is reduced with respect to
the above rules.
As a warm-up, we first begin with the simple case where each element
of U is contained in at most b sets of F. A special case of this situation is
Max Cut where b = 2.
Lemma 5.2. If each element e ∈ U occurs in at most b sets of F then the
Unique Coverage problem admits a kernel of size b(k − 1).
Proof.
Find a maximal collection F ′ of pairwise disjoint
S sets in F. If
S
| S∈F ′ S| ≥ k, we are done. Therefore assume that | S∈F ′ S| ≤ k − 1.
Since every set in F \ F ′ intersects some set in F ′ , and every element of U
occurs in at most b sets in F, we have |F \F ′ | ≤ (k−1)(b−1). But |F ′ | ≤ k−1
and so |F| ≤ b(k − 1).
5.4.1
Bounded Intersection Size
Now consider the situation where for all Si , Sj ∈ F, i 6= j, we have |Si ∩
Sj | ≤ c, for some constant c. In this case we say that the problem instance
has intersection size bounded by c and show that the problem admits a
polynomial kernel of size O(kc+1 ). First consider the case when |Si ∩Sj | ≤ 1.
Lemma 5.3. Suppose that for all Si , Sj ∈ F, i 6= j, we have |Si ∩ Sj | ≤ 1.
If an element e ∈ U is covered by at least k + 1 sets, then one can obtain a
solution covering k elements uniquely in polynomial time.
Proof. Suppose an element e ∈ U is covered by the sets S1 , . . . , Sk+1 . Then
by reduction rule R2, at most one of these sets can have size 1. The remaining k sets uniquely cover at least one element each.
One can now easily obtain a kernel of size k2 for the case when the
intersection size is at most 1.
Lemma 5.4. Suppose that for all Si , Sj ∈ F, i 6= j, we have |Si ∩ Sj | ≤ 1.
If |F| ≥ k2 , then there exists T ⊆ F that covers at least k elements uniquely.
Proof. If an element appears in at least k + 1 sets then we are done by
Lemma 5.3. Otherwise every element appears in at most k sets and by
Lemma 5.2 we have a kernel of size k(k − 1) < k 2 .
5.4. Unique Coverage: The Standard Version
83
Next, we generalize these observations to the case when |Si ∩ Sj | ≤ c, for
some constant c.
Theorem 5.3. Suppose that for all Si , Sj ∈ F, i 6= j, we have |Si ∩ Sj | ≤ c,
for some positive constant c. If |F| ≥ kc+1 then one can, in polynomial
time, find a subset T ⊆ F that covers k elements uniquely.
Proof. By induction on c. For c = 1, this follows from Lemma 5.4. Assume
the theorem to hold for c ≥ 1. Let c ≥ 2. GreedilySobtain a maximal
collection F ′ = {S1 , . . . , Sp } of pairwise
disjoint sets. If | Si ∈F ′ Si | ≥ k then
S
we are done. Therefore assume | S∈F ′ S| ≤ k − 1 (this also implies p ≤ k −
1). Since |F| ≥ kc+1 , and
S since every set in F intersects with at least one set
in F ′ , there exists e ∈ S∈F ′ S such that at least kc +1 sets in F \{S1 , . . . , Sp }
contain e. For otherwise, |F| ≤ (k − 1)kc + p < kc+1 , a contradiction.
Let T1 , . . . , Tkc +1 be some kc + 1 such sets. Delete e from each of these sets.
We obtain at least kc nonempty distinct sets T1′ , . . . , Tk′ c (there is at most one
set consisting only of the element e which is deleted in this process). Note
that any two of these sets intersect in at most c − 1 elements. By induction
hypothesis, there exists a collection T ′ ⊆ {T1′ , . . . , Tk′ c } that uniquely covers
at least k elements, and thus there exists a collection T ⊆ {T1 , . . . , Tkc } that
uniquely covers at least k elements (just take the solution for T ′ and add e
to every set in it). This proves the theorem.
Corollary 5.1. Unique Coverage admits a kernel of size kc+1 in the case
where any two sets in the family intersect in at most c elements.
By reduction rule R1 we have c ≤ k − 1 and therefore for the general
case we have a kernel of size kk .
Corollary 5.2. Unique Coverage is fixed-parameter tractable and admits
a problem kernel of size kk .
An algorithm that checks all possible subsets of a family of size kk to see
whether any of them uniquely covers at least k elements is an FPT-algorithm
k
with time complexity O ∗ (2(k ) ). But note that we can assume without loss of
generality that every set in the solution covers at least one element uniquely.
Thus it suffices to check whether subfamilies of size at most k uniquely
2
2
cover at least k elements. This can be done in time O ∗ (kk ) = O ∗ (2k log k ).
However, it turns out that a much better kernel can be obtained for the
general case.
5.4.2
A Better Kernel for the General Case
We now show that Unique Coverage has a kernel of size 4k using a
result on strong systems of distinct representatives. Given a family of
sets F = {S1 , . . . , Sm }, a system of distinct representatives for F is an
84
Chapter 5. The Unique Coverage Problem
m-tuple (x1 , . . . , xm ) where the elements xi are distinct and xi ∈ Si for
all i = 1, 2, . . . , m. Such a system is strong if we additionally have xi ∈
/ Sj
for all i 6= j.
Theorem 5.4 ([83]). In any family of more than r+s
sets of cardinality
s
at most r, at least s + 2 of its members have a strong system of distinct
representatives.
Given an instance (U , F, k) of Unique Coverage, put r = k−1 and s =
k in the statement of the above theorem and we have a kernel of size 2k−1
k−1 ≤
2k 2k
k ≤2 .
Corollary 5.3. Unique Coverage admits a problem kernel of size 4k .
As noted before, this is essentially the best possible kernel for this problem since Dom et al. have shown that Unique Coverage does not admit
a kernel of size polynomial in k unless the Polynomial Hierarchy collapses
to the third level [46].
2
Corollary 5.4. There is an O ∗ (4k ) time algorithm for the Unique Coverage problem.
Proof. A subfamily that covers k elements uniquely has size at most k.
Therefore it is sufficient to consider all possible size-k subfamilies of the
2
4k -kernel. This takes time O∗ (4k ).
In Section 5.6, we provide a better algorithm for Unique Coverage
with running time O(2O(k log log k) · mk + m2 ), where m is the number of sets
in the input family. At this point, we note that Gen Unique Coverage
(see Section 5.3) is fixed-parameter tractable. Recall the problem definition:
Given (U , F) and nonnegative integers k and t, does there exist a subfamily
of F that covers k elements at least once and at most t times? Here k is the
parameter. An instance (U , F, k, t) is trivially a yes-instance if |F| > 4k .
Otherwise |F| ≤ 4k and we have a kernel.
Corollary 5.5. Gen Unique Coverage is fixed-parameter tractable.
For the case where each set of the input family has size at most b, for
some constant
b, there is a better kernel. By Theorem 5.4, if there exists at
least b+k
sets
in the input family, then there exists at least k sets with a
k
strong system of distinct representatives.
Corollary 5.6. If each set S ∈ F has size at most b then the Unique
Coverage problem has a kernel of size O(2b+k ).
5.5. Budgeted Unique Coverage
5.5
85
Budgeted Unique Coverage
In this section we consider the Budgeted Unique Coverage problem
where each set in the input family has a cost and each element in the universe
has a profit; the goal is to decide whether there exists a subfamily of total
cost at most B that uniquely covers elements of total profit at least k. By
parameterizing on k or B or both we obtain different parameterized versions
of this decision question.
5.5.1
Intractable Parameterized Versions
We first consider the Budgeted Max Cut problem which is a specialization of the Budgeted Unique Coverage problem. An instance of this
problem is an undirected graph G = (V, E) with a cost function c : V → Q+
on the vertex set and a profit function p : E → Q+ on the edge set; positive
rational numbers B and k. The question is whether there exists a cut (T, T ′ )
such that the total cost of the vertices in T is at most B and the total profit
of the edges crossing the cut is at least k.
We first show that the Budgeted Max Cut problem with arbitrary
positive rational costs and profits is probably not in FPT.
Lemma 5.5. The Budgeted Max Cut problem with arbitrary positive
rational costs and profits with parameters B and k is not in FPT, unless P =
NP.
Proof. Suppose there exists an algorithm for Budgeted Max Cut (with
arbitrary positive costs and profits) with running time O(f (k, B) · p(n)),
where p is a polynomial in n. We will use this to solve the decision version
of Max Cut in polynomial time. Let (G = (V, E), k) be an instance of
the Max Cut problem, where |V | = n. Assign each vertex of the input
graph cost 1/n and each edge profit 1/k. Let the budget B = 1/2 and the
profit k′ = 1. Clearly, G has a maximum cut of size at least k if and only if
there exists S ⊆ V of total cost at most B such that the total profit of the
edges crossing the cut (S, V \ S) is at least k′ . And this can be answered in
time O(f (1, 1/2) · p(|V |)), implying P = NP.
Theorem 5.5. The Budgeted Unique Coverage problem with arbitrary
positive rational costs and profits is not in FPT, unless P = NP.
We next show that the Budgeted Max Cut problem parameterized
by the budget B alone is W[1]-hard even when the costs and profits are
positive integers.
Lemma 5.6. The Budgeted Max Cut problem with positive integer costs
and profits, parameterized by the budget B is W[1]-hard.
86
Chapter 5. The Unique Coverage Problem
Proof. To show W[1]-hardness, we exhibit a fixed-parameter reduction from
the Independent Set problem to the Budgeted Max Cut problem with
unit costs and profits. Let (G = (V, E), B) be an instance of Independent
Set with |V | = n. For every vertex u ∈ V add |V | − 1 − deg(u) new vertices
and connect them to u. Call the resulting graph G′ . Define c(v) = 1 for
all v ∈ V (G′ ) and p(e) = 1 for all e ∈ E(G′ ). Note that every vertex u ∈ G
has degree |V | − 1 in G′ . We let (G′ = (V ′ , E ′ ), B, k = B(n − 1)) be the
instance of Budgeted Max Cut.
Claim. The graph G has an independent set of size B if and only if G′ has a
cut (S, V ′ \ S) such that |S| = B and at least k = B(n − 1) edges lie across
it.
If G has an independent set S of size B, then clearly S is independent
in G′ . The cut (S, V ′ \ S) does indeed have B(n − 1) edges crossing it, as
every vertex of S has degree n−1. Next suppose that G′ has a cut (S, V ′ \S)
with B(n − 1) edges crossing it such that |S| = B. Note that every vertex
in S must be a vertex from G. Otherwise the cut cannot have B(n−1) edges
crossing it. Suppose two vertices u and v in S are adjacent. Then both u
and v contribute less than n − 1 edges to the cut. Since each vertex in S
contributes at most n − 1 edges to the cut, the number of edges crossing the
cut must be less than B(n − 1), a contradiction. Hence S is independent
in G′ and and G has an independent set of size B.
Since the Budgeted Unique Coverage problem is a generalization
of Budgeted Max Cut we have
Theorem 5.6. The Budgeted Unique Coverage problem with positive
integer costs and profits, parameterized by the budget B is W[1]-hard.
5.5.2
Parameterizing by both B and k
We now assume that, unless otherwise mentioned, both costs and profits are
positive integers and that both B and k are parameters. Let (U , F, c, p, B, k)
be an instance of Budgeted Unique Coverage. We may assume that
for all Si , Sj ∈ F, i 6= j, we have
1. Si 6= Sj ;
2. c(Si ) ≤ B;
3. p(Si ) ≤ k − 1;
4. B ≥ 2.
For if c(Si ) > B then Si cannot be part of any solution and may be discarded;
if p(Si ) ≥ k then the given instance is trivially a yes-instance. Note that
5.5. Budgeted Unique Coverage
87
the condition p(Si ) ≤ k − 1 implies that |Si | ≤ k − 1. Also observe that if
the first three conditions hold, then in a yes-instance we must have B ≥ 2.
For some of the results in this section, we need the following lemma.
Lemma 5.7. For all t ≥ 2k,
Proof. We show that
t
t−k
t
t−k
t
t
t−k t
t
≥ (2e)−k .
≤ (2e)k for all t ≥ 2k.
t−k k
k
k
=
1+
· 1+
t−k
t−k
k
k
≤ ek · 1 +
t−k
≤ ek · 2k = (2e)k
The last inequality follows since t ≥ 2k.
Demaine et al. [43] show that there exists an Ω(1/ log n)-approximation
algorithm for Budgeted Unique Coverage (Theorem 4.1). We use the
same proof technique to show the following.
Lemma 5.8. Let (U , F, c, p, B, k) be an instance of Budgeted Unique
Coverage and let c : F → Q≥1 and p : U → Q≥1 . Then either
1. one can find, in polynomial time, a subfamily F ′ ⊆ F with total cost
at most B such that the total profit of elements uniquely covered by F ′
is at least k; or
2. for every subfamily H with total cost at most B, we have |
18k log B.
S
S∈H S|
≤
Proof. Let H = {S1 , . . . , Sr } be a subfamily
of F with budget at most B
S
which maximizes |U ′ |, where U ′ = S∈H S. For u ∈ U , let fu denote the
number of sets of H containing u. Partition U ′ into sets C0 , C1 . . . Ct−1 , such
that u ∈ Ci if 2i ≤ fu ≤ 2i+1 − 1. Note that i ranges from 0 to log(r + 1) − 1,
since the frequency of any element in Ct−1 is at most 2t − 1, which in turn
is at most r. So t ≤ log(r + 1).
Clearly there exists j such that |Cj | ≥ |U ′ |/ log(r + 1). Fix j to be
an index for which |Cj | ≥ |U ′ |/ log(r + 1). Fix u ∈ Cj and note that
2j ≤ fu ≤ 2j+1 − 1. Construct a subfamily F ′ from H by going through
each set in H and including it in F ′ with probability 1/2j+1 . Denote by lu
88
Chapter 5. The Unique Coverage Problem
the probability that u is covered uniquely by F ′ . Then
(fu −1)
fu
1
1 − 2j+1
lu =
j+1
2
(fu −1)
1
≥ 21 1 − 2j+1
(since fu ≥ 2j ).
(2j+1 −2)
1
≥ 21 1 − 2j+1
(since fu ≤ 2j+1 − 1).
(2j+1 )
1
≥ 12 1 − 2j+1
≥
1
4e .
(by Lemma 5.7).
Let Xu be an indicator random variable which takes value 1P
if u is covered
uniquely by the subfamily F ′ , and 0 otherwise. Also, let X = u Xu . Then
E(X) =
X
u
E(Xu ) ≥
X
u∈Cj
E(Xu ) =
X
u∈Cj
lu ≥
|U ′ |
.
4e log(r + 1)
Let the set of elements uniquely covered by F ′ be Q. Then the total profit
of these uniquely covered elements is at least |Q| and
E(|Q|) ≥
|U ′ |
1
·
,
4e log(r + 1)
as every uniquely covered element contributes at least 1 to the total profit.
Since r is bounded above by B, the total expected profit of these elements
is at least
1
|U ′ |
·
.
(5.1)
4e log(B + 1)
This implies that if k is at most the quantity in expression 5.1 then the total
profit of uniquely covered elements of subfamily F ′ is at least k; else, we
have |U ′ | ≤ 4ek log(B + 1) ≤ 12k log B.
We can design an algorithm that finds a subfamily of total budget at
most B that uniquely covers elements with total profit at least k as follows.
Define p′ to be the unit profit function, that is, p′ (u) = 1 for all u ∈ U .
We find a subfamily H′ in place of H using an approximation algorithm for
the Maximum Coverage problem (where the objective is to maximize the
elements covered by the subfamily). To do this we run the polynomial-time
(1 − 1/e)-ratio approximate algorithm described in [89] for the Budgeted
Maximum Coverage problem with universe U , family F, cost function c,
profit function p′ and budget B. This returns a subfamily H′ of total cost
at most B that covers at least (1 − 1/e) · opt = d · opt elements, where opt
denotes the maximum number of elements in any family of total cost at
most B. From H′ one can deterministically obtain a subfamily F ′ that
uniquely covers elements with total profit at least
d · opt
1
·
4e log(B + 1)
5.5. Budgeted Unique Coverage
89
by the method of conditional probabilities (see [104]).
Hence depending on the value of k one can, in polynomial time, either
obtain a subfamily F ′ ⊆ F with budget at most B such that the total profit
of uniquely covered elements is at least k; or every subfamily H with budget
4e
· k log(B + 1) ≤ 18k log B elements.
at most B contains at most 1−1/e
The first step of our algorithm is to apply Step 1 of Lemma 5.8. Therefore
from now on we assume that every subfamily of total cost at most B covers
at most 18k log B elements of the universe.
We now proceed to show that Budgeted Unique Coverage is in FPT
by an application of the color-coding technique. We first show this for the
case when the costs and profits are all one and then handle the more general
case of integral costs and profits. Therefore let (U , F, B, k) be an instance of
Budgeted Unique Coverage with unit costs and profits. For this version
of the problem, we have to decide whether there exists a subfamily F ′ ⊆ F
of size at most B that uniquely covers at least k elements. A subfamily F ′
of size at most B that uniquely covers at least k elements is called a solution
subfamily.
To develop our color-coding algorithm, we use two sets of colors Cg and Cb
with the understanding that the (good) colors from Cg are used for the
elements that are uniquely covered and the (bad) colors from Cb are used
for the remaining elements. In the present setting, Cg = {1, . . . , k} and Cb =
{k+1}. Note that for our algorithms, any subset of k colors can play the role
of good colors. For ease of presentation, we fix a set of good and bad colors
while describing our randomized algorithms. Our derandomized algorithms
assume that any set of k colors may be good colors.
We now describe the notion of a good configuration. Given a function h : U → Cg ⊎ Cb and F ′ ⊆ F, define U (F ′ ) to be the set of elements
covered (not necessarily uniquely) by F ′ and h(F ′ ) to be the set of colors
assigned to the elements in U (F ′ ). That is,
U (F ′ ) :=
[
S∈F ′
S;
h(F ′ ) :=
[
i∈U (F ′ )
{h(i)}.
Definition 5.1. Given h : U → Cg ⊎ Cb and Cg′ ⊆ Cg , we say that F ′ ⊆ F
has a good configuration with respect to (w.r.t.) h and Cg′ if
1. h(F ′ ) ∩ Cg = Cg′ , and
2. there are exactly |Cg′ | elements in F ′ that are assigned colors from Cg′
and these elements are uniquely covered by F ′ .
We also say that F has a good configuration w.r.t. h and Cg′ if there exists
a subfamily F ′ with a good configuration w.r.t. h and Cg′ . Call F ′ a witness
subfamily.
90
Chapter 5. The Unique Coverage Problem
A solution subfamily (for the unit costs and profits version) is a subfamily F ′ ⊆ F with at most B sets and which uniquely covers at least k
elements.
The next lemma shows that if (U , F, B, k) is a yes-instance of Budgeted Unique Coverage with unit costs and profits, and h is chosen
uniformly at random from the space of all functions from U to [k + 1], then
a solution subfamily F ′ has a good configuration w.r.t. h and Cg with high
probability. Note that such a uniformly chosen h maps every element from U
uniformly at random to an element in [k + 1].
Lemma 5.9. Let (U , F, B, k) be a yes-instance of Budgeted Unique
Coverage with unit costs and profits and let h : U → [k + 1] be a function
chosen uniformly at random. Then a solution subfamily F ′ has a good configuration w.r.t. h and Cg with probability at least 2−k(18 log B log(k+1)−log k+log e) .
Proof. Let F ′ be a solution subfamily with at most B sets that covers the
elements Q = {i1 , . . . , ik } uniquely. Then p := |U (F ′ )| ≤ 18k log B. To
complete the proof, we show that F ′ has a good configuration with respect
to h and Cg with probability at least 2−k(18 log B log(k+1)−log k+log e) . For F ′ to
have a good configuration, we must have h(i) = k + 1 for all i ∈ U (F ′ ) \ Q
and h(i1 ), . . . , h(ik ) a permutation of 1, . . . , k. The probability Pr that this
happens is:
k!
1
′ )\Q| ×
|U
(F
(k + 1)k
(k + 1)
k
1
k
= ek ln(k/e)−p ln(k+1)
≥
e
(k + 1)p
Pr =
≥ ek ln(k/e)−18k log B ln(k+1)
= 2−k(18 log B log(k+1)−log k+log e)
This proves the lemma.
Given a coloring h, how do we find out whether F has a good configuration w.r.t. h and Cg ? We answer this next.
Finding a good configuration. Observe that if F has a good configuration
w.r.t. h and Cg , then any witness subfamily F ′ covers at least k elements
uniquely. To locate such a family of size at most B we use dynamic programming over subsets of Cg . To this end, let W be a 2k × B array where we
identify the rows of W with subsets of Cg and the columns with the size of a
subfamily. For a fixed coloring function h, a subset Cg′ ⊆ Cg and 1 ≤ i ≤ B,
define W [Cg′ ][i] as follows:

if there exists F ′ ⊆ F, with |F ′ | ≤ i, with a
 1,
′
good configuration w.r.t. Cg′ and h.
W [Cg ][i] =

0,
otherwise.
5.5. Budgeted Unique Coverage
91
The entry corresponding to W [∅][i] is set to 1 for all 1 ≤ i ≤ B, as a
convention. We fill this array in increasing order of the sizes of subsets of Cg .
Let T be the family of all sets S ∈ F such that g(S) := h(S) ∩ Cg ⊆ Cg′ and
for each c ∈ g(S) there exists exactly one element e ∈ S with h(e) = c. Then
_
W [Cg′ ][i] =
W [Cg′ \ g(S)][i − 1].
S∈T
The correctness of the algorithm is immediate. Clearly if W [Cg ][B] = 1,
then a subfamily with at most B sets that uniquely covers at least k elements
exists, and can be found out by simply storing the witness families F ′ for
every entry in the table and backtracking. The time taken by the algorithm
is O(2k Bmk), since the size of the array is 2k B and each entry of the array
can be filled in time O(mk), where m = |F|.
Lemma 5.10. Let (U , F, B, k) be an instance of Budgeted Unique Coverage with unit costs and profits and h : U → C a coloring function. Then
one can find a subfamily F ′ of size at most B with a good configuration
w.r.t. h and Cg , if there exists one, in time O(2k Bmk).
A randomized algorithm for Budgeted Unique Coverage with unit
costs and profits is as follows.
1. Randomly choose a coloring function h : U → {1, . . . , k + 1}.
2. Apply Lemma 5.10 and check whether there exists a family F ′ of size
at most B that is witness to a good configuration w.r.t. h and Cg . If
such a family exists, return yes, else go to Step 1.
By Lemma 5.9, if the given instance is a yes-instance, the probability
that a solution subfamily F ′ has a good configuration w.r.t. a randomly
chosen function h : U → C and Cg is at least 2−k(18 log B log(k+1)−log k+log e) .
By Lemma 5.10, one can find such a subfamily in time O(2k Bmk).
Theorem 5.7. Let (U , F, B, k) be an instance of Budgeted Unique Coverage with unit costs and profits. There exists a randomized algorithm that
finds a subfamily F ′ of size at most B covering at least k elements uniquely,
if there exists one, in expected O(218k log B log(k+1) · Bmk) time.
Improving the running time
It is clear that if a solution subfamily F ′ is to have a good configuration
w.r.t. a randomly chosen coloring function h and Cg , then h must assign
all the non-uniquely covered elements of F ′ the color in Cb . Intuitively, if
we increase the number of colors in Cb , we increase the probability that a
specific target subfamily has a good configuration w.r.t. a randomly chosen
coloring function. We formalize this intuition below.
92
Chapter 5. The Unique Coverage Problem
Lemma 5.11. Let (U , F, B, k) be a yes-instance of Budgeted Unique
Coverage with unit costs and profits; let Cg = [k], Cb = {k + 1, . . . , q}
and C = [q] so that q ≥ 2k. If h : U → C is chosen uniformly at random
then every solution subfamily F ′ with p elements of the universe has a good
k
− kp
k
−k
configuration w.r.t. h and Cg with probability at least e
(2e) q .
q−k
Proof. Let the set of elements uniquely covered by F ′ be Q = {i1 , . . . , ik }.
For F ′ to have a good configuration, the function h must map every element
of U (F ′ ) \ Q to Cb and map Q to Cg injectively. Therefore the probability Pr
that F ′ has a good configuration w.r.t. Cg and a randomly chosen h is:
k!
(q − k)p−k
× k
p−k
q
q
p k
1
q−k
·
· kk e−k
≥
q
q−k
k k k p
· 1−
≥ e−k ·
q−k
q
k k
kp
−
≥ e−k ·
(by Lemma 5.7).
· (2e) q
q−k
Pr =
This proves the lemma.
If (U , F, B, k) is a yes-instance of Budgeted Unique Coverage with
unit costs and profits then p ≤ 18k log B. As we observed earlier, we
have B ≥ 2. By Lemma 5.11, a solution subfamily F ′ has a good configuration w.r.t. a randomly chosen coloring function h and Cg with probability
−k · (2e)−kp/q . Setting q = k + p, this expression works
at least e−k · ( q−k
k )
out to be:
k
k
Pr ≥ e−k ·
· (2e)−kp/(k+p)
p
−k log e−k log
kp
p
− k+p
·log 2e
k
−k log e−k log
p
k
− 1+k/p
·log 2e
k
≥ 2
= 2
p
≥ 2−k log e−k log k −k log 2e .
Now setting p = 18k log B this expression works out to 2−8.1k−k log log B .
Combining this with Lemma 5.10, we obtain:
Theorem 5.8. Let (U , F, B, k) be an instance of Budgeted Unique Coverage with unit costs and profits. Then one can find a subfamily F ′ of
size at most B covering at least k elements uniquely, if there exists one, in
O(29.1k+k log log B · Bmk) expected time.
5.5. Budgeted Unique Coverage
93
Derandomization
We now discuss how to derandomize the algorithms described in the last
subsection. In general, randomized algorithms based on the color-coding
method are derandomized using a suitable family of hash functions or “universal sets”. Fix integers k, s, t, n, where [n] is the universe, [t] is the set of
colors, s and k denote, respectively, the size of a solution subfamily and the
number of elements uniquely covered by it. Note that t ≥ k + 1 and, by
Lemma 5.8, s = ⌈18k log B⌉. We need a family of functions from U = [n]
to [t] such that for all S ⊆ U of size s and all X ⊆ S of size k, there exists a
function h in the family which maps X injectively and the colors it assigns
to the elements in S \ X are different from the ones it assigns to those in X.
See Figure 5.1
U
S
X
h
C
1−1
Cg
Cb
Figure 5.1: The sets in the definition of a (k, s)-hash family. We assume U = [n],
C = [t], |S| = s and |X| = k.
Such hash families are called (k, s)-hash families (with domain [n] and
range [t]) and they were introduced by Barg et al. [14] in the context of
particular class of codes called parent identifying codes. At this point, we
recall the definition of an (n, t, s)-perfect hash family.
Definition 5.2 ((n, t, s)-Perfect Hash Family). A family H of functions
from [n] to [t] is called an (n, t, s)-perfect hash family if for every subset X ⊆
[n] of size s, there is a function h ∈ H that maps X injectively.
Note that an (n, t, s)-perfect hash family is a (k, s)-hash family with domain [n] and range [t] for any k ≤ s, and a (k, s)-hash family with domain [n]
and range [t] is an (n, t, k)-perfect hash family. Therefore (k, s)-hash families
may be thought of as standing in between (n, t, k)-perfect and (n, t, s)-perfect
hash families.
Our deterministic algorithm simply uses functions from these families H
for coloring and is described in Figure 5.2. Given an instance (U , F, B, k) of
Budgeted Unique Coverage with unit costs and profits, we let n = |U |,
C = [t], and s to be the closest integer to our estimate in Lemma 5.8,
94
Chapter 5. The Unique Coverage Problem
for each h ∈ H do
for each subset X ⊆ C = [t] of size k do
1. Define Cg = X and Cb = C \ X;
2. Apply Lemma 5.10 and check whether there exists a subfamily F ′ of size at most B which has a good configuration
w.r.t. Cg and h;
3. if yes, then return the corresponding F ′ ;
return no;
Figure 5.2: Deterministic algorithm for Budgeted Unique Coverage.
which is O(k log B). The correctness of the algorithm follows from the
description—if a witness subfamily for the given F exists, at least one h ∈ H
will color all the uniquely covered elements of the witness subfamily distinctly, thereby resulting
in a good
configuration. The running time of the
t
k
algorithm is O |H| · k · 2 Bmk .
Alon et al. [2] provide explicit constructions of (k, s)-hash families when
the range is k + 1 and ks, respectively.
Theorem 5.9 (Alon et al. [2]). Let 2 ≤ k < s. There is an explicit construction of a (k, s)-hash family H with domain [n] and
• range [k + 1] of size at most 2ck log s · logk+1 n, for some absolute constant c > 0;
• range [ks] of size O(k2 s2 log n).
If t = k + 1, then by the above theorem, the running time of our deterministic algorithm is O(2O(k log k+k log log B) · Bmk · log n); when t = ks, the
running time works out to be O(2O(k log k+k log log B) · mk5 · B log2 B · log n).
Theorem 5.10. Let (U , F, B, k) an instance of the Budgeted Unique
Coverage problem with unit costs and profits. Then one can find a subfamily F ′ of size at most B covering at least k elements uniquely, if there
exists one, in time O(2O(k log k+k log log B) · Bmk · log n).
We next give alternate running time bounds using standard (n, t, s)perfect hash families for derandomizing our algorithm.
Theorem 5.11 ([6, 119, 32]). There exist explicit constructions of (n, t, s)perfect hash families of size
• 2O(s) log n when t = s; and
• sO(1) log n when t = s2 .
5.5. Budgeted Unique Coverage
95
In fact, when t = s, an explicit construction of an s-perfect hash family of
size 6.4s log2 n in time 6.4s n log2 n is known.
For t = s, using the construction of s-perfect hash
families by Chen et
al. [32], we obtain a running time of O(6.4s log2 n · ks · 2k · Bkm). Since s =
O(k log B), this expression simplifies to O(2O(k log B) · log2 n · Bmk). For t =
s2 , we can use a hash family of size sO(1) log n [6], and the expression for the
running time then works out to be O(2O(k log k+k log log B) · log n · Bmk). We
thus have
Theorem 5.12. Let (U , F, B, k) be an instance of the Budgeted Unique
Coverage problem with unit costs and profits. Then one can find a subfamily F ′ of size at most B covering at least k elements uniquely, if there
exists one, in time O(min{2O(k log B) , 2O(k log k+k log log B) } · log2 n · Bmk).
In general, there is no relation between the parameters k and B, but
if B is small or nearly equal to k then the running time in Theorem 5.12 is
better than that in Theorem 5.10.
We now consider existential results concerning hash families. The following is known about (n, t, s)-hash families.
Theorem 5.13 ([101]). For all positive integers n ≥ t ≥ s ≥ 2, there exists
2
an (n, t, s)-perfect hash family ∆(n, t, s) of size es /t s ln n.
Alon et al. [2] provide existential bounds for (k, s)-hash families when the
range t = k + 1 and one such bound (see Theorem 3 in [2]) is the following.
If H is a (k, s)-hash family with a domain [n] and range [k + 1] then
logk+1 n
k!(s − k)s−k
= s
− o(1),
|H|
s (s − 1) ln(k + 1)
which implies that
|H| ≤
ss (s − 1) ln(k + 1) log k+1 n
.
k!(s − k)s−k
If we assume that s ≥ 2k, then using Lemma 5.7, this expression can be
bounded above by (2e)k · sk+1 ln(k + 1) log k+1 n. As stated in Theorem 5.9,
Alon et al. [2] describe explicit constructions of such (k, s)-hash family of
size sck · logk+1 n which is still exponentially larger than the existential
bound.
In the lemmas that follow, we provide existential bounds for an arbitrary
range.
Lemma 5.12. Let k ≤ s ≤ n be positive integers and let t ≥ 2k be an
integer. There exists a (k, s)-hash family H with domain [n] and range [t]
of size ek (2e)sk/t · s log n.
96
Chapter 5. The Unique Coverage Problem
Proof. Let A = {h : [n] → [t]} be the set of all functions from [n] to [t].
For h ∈ A, S ⊆ [n] of size s and X ⊆ S of size k, define h to be an (X, S)hash function if h maps X injectively such that h(X) ∩ h(S \ X) = ∅ and
not an (X, S)-hash function otherwise.
Fix S ⊆ [n] of size s and X ⊆ S of size k. The probability Pr that a
function h picked uniformly at random from A is an (X, S)-hash function,
is given by:
t
s−k
k k!(t − k)
Pr =
ts
k k
t
k
t − k s−k
1
>
·
· k·
k
e
t
t
s−k
1 t−k
= k
e
t
k(s−k)/t
1
1
≥ k·
(By Lemma 5.7.)
e
2e
ks/t
1
1
≥ k·
.
e
2e
The probability that the function h is not an (X, S)-hash function is less
than 1 − e−k (2e)−ks/t . If we pick N functions uniformly at random from A
then the probability that none of these functions is an (X, S)-hash function
is less than (1 − e−k (2e)−ks/t) )N . The probability that none of these N
functions is an (X, S)-hash function for some (S, X) pair is less than
n
s
(1 − e−k (2e)−ks/t )N ,
s
k
which in turn is less than ns (1 − e−k (2e)−ks/t )N . For this family of N
functions to contain an (X, S)-hash function for every (S, X) pair, we would
want
ns (1 − e−k (2e)−ks/t )N ≤ 1.
(5.2)
Inequality 5.2 implies that
s ln n + N ln(1 − e−k (2e)−ks/t ) ≤ 0.
One can show that given any ǫ ∈ (0, 1) there exists cǫ > 0 such that for
all x ∈ (0, ǫ), we have −cǫ x ≤ ln(1 − x). In fact, one may choose cǫ =
(| ln(1 − ǫ)| + 1)/ǫ. For k and s sufficiently large we have e−k (2e)−ks/t < 1/4
and we may choose ǫ = 1/4. Then cǫ ≈ 5.2 and we have:
s ln n − 5.2N · e−k (2e)−sk/t ≤ 0.
which shows that N ≥ ek (2e)ks/t · s log n.
5.5. Budgeted Unique Coverage
97
Lemma 5.13. Let k ≤ s ≤ n be positive integers and let t ≥ k + 1.
Then there exists a (k, s)-hash family with domain [n] and range [t] of
size 2O(k log(s/k)) · s log n .
Proof. Let F = ∆(n, m, s), the (n, m, s)-perfect hash family obtained from
Theorem 5.13, where we set m = ⌈s2 /(k log(s/k))⌉. Let G be a family
of functions gX from [m] to [t], indexed by k-element subsets X of [m] as
follows. The function gX maps X in an one-one, onto fashion to {1, . . . , k}
and maps an element of [m] \ X to an arbitrary element in {k + 1, . . . , t}.
Our required family T of functions from [n] to [t] is obtained by composing
the families F and G. It is easy to see that T is an (k, s)-hash family and
has the claimed bound for its size.
Note that Lemma 5.12 requires that t ≥ 2k and that for Lemma 5.13 we
have no restriction on t. Also note that the bound in Lemma 5.13 is existential as it uses the existential bound of Theorem 5.13. If we had an explicit
construction of a (k, s)-hash family satisfying the bound in Lemma 5.12,
then by setting s = t = O(k log B), we would have obtained a running time
of O(2O(k log log B) · k log B log n) which is significantly better than that given
in Theorem 5.12. We believe that this is motivation for studying explicit
constructions of (k, s)-hash families for an arbitrary range.
Generalized Costs and Profits
Observe that the algorithms for Budgeted Unique Coverage with unit
costs and profits had two components—one component ensured that the
probability that a solution subfamily has a good configuration with respect
to a random coloring h and Cg is sufficiently high (see Lemma 5.9), and
the other had to do with finding a witness subfamily given a coloring (see
Lemma 5.10). Note that Lemma 5.8 and the discussions proceeding Theorem 5.8 continue to hold when either the cost or the profit is integral. The
derandomization procedures given in the last subsection also go through for
these general cases. We now show how to modify the dynamic programming
algorithm when the costs and profits are arbitrary integers.
Dynamic programming with integral costs and profits Consider an instance (U , F, B, k) of the problem with cost function c and profit function p.
Recall that the colors used are from C = Cg ∪ Cb and that we are given a
coloring function h. As before, we define an array W of size 2k × B and
associate the rows with subsets of Cg and columns with the cost of a subfamily. For a subfamily H, let p(H) denote the total profit of elements uniquely
covered by H. For Cg′ ⊆ Cg and 1 ≤ i ≤ B, let Wi [Cg′ ] denote the set of
all subfamilies of budget at most i which has a good configuration w.r.t. h
and Cg′ .
98
Chapter 5. The Unique Coverage Problem
For Cg′ ⊆ Cg and 1 ≤ i ≤ B, define W [Cg′ ][i] as follows:
W [Cg′ ][i]
=
maxH∈Wi [Cg′ ] {p(H)}, if Wi [Cg′ ] 6= ∅;
0,
otherwise.
The entry corresponding to W [∅][i] is set to 0 for all 1 ≤ i ≤ B as a
convention. We fill this array in increasing order of sizes of the subsets of Cg .
For S ∈ F, define g(S) = h(S) ∩ Cg to be the set of good colors assigned
to elements in S and p(S) to be the total profit of elements in S assigned
colors from g(S). Let T be the subfamily containing all sets S ∈ F such
that g(S) ⊆ Cg′ and for each c ∈ g(S) there exists exactly one element e ∈ S
with h(e) = c. Then
W [Cg′ ][i] = max p(S) + W [Cg′ \ g(S)][i − c(S)]
S∈T
The correctness of the algorithm is immediate. Clearly, if W [Cg ][B] ≥ k,
then there exists a subfamily with total cost at most B which uniquely covers
elements with total profit at least k. Such a family can be found by simply
storing the witness set for every entry in the table and backtracking. The
time taken by the algorithm is O(2k Bmk + |C|) = O(2k Bmk), where m =
|F|.
Lemma 5.14. Let (U , F, B, k, c, p) be an instance of Budgeted Unique
Coverage with integral costs and profits, and h : U → C be a coloring
function. Then one can find a subfamily H of total cost at most B which
has a good configuration w.r.t. coloring h and Cg , if there exists one, in time
O(2k Bmk).
Note that Lemma 5.14 holds even when profits are from Q≥1 . When the
costs are from Q≥1 and profits are positive integers then we can modify the
dynamic programming algorithm as follows. Define an array W of size 2k ×k
and identify its rows with subsets of Cg and columns with the profit of a
subfamily. For Cg′ ⊆ Cg and 1 ≤ i ≤ k, define Wi [Cg′ ] and W [Cg′ ][i] as follows:
Wi [Cg′ ] denotes the set of all subfamilies with profit at least i and which have
a good configuration w.r.t. the given coloring function h and Cg′ ; and
W [Cg′ ][i]
=
minH∈Wi [Cg′ ] {c(H)}, if Wi [Cg′ ] 6= ∅;
∞,
otherwise.
The entry W [∅][i] is set to 0 for all 1 ≤ i ≤ k. Any entry of the form W [Cg′ ][i]
where Cg′ ⊆ Cg and i ≤ 1 is identified with the entry W [Cg′ ][1]. Given Cg′ ⊆ Cg ,
let T be the subfamily containing all sets S ∈ F such that g(S) := h(S) ∩
Cg ⊆ Cg′ and for each c ∈ g(S) there exists exactly one e ∈ S with h(e) = c.
Then
W [Cg′ ][i] = min{c(S) + W [Cg′ \ g(S)][i − p(S)]}.
S∈T
5.6. Algorithms for Special Cases
99
If W [Cg ][k] ≤ B then there exists a subfamily with total cost at most B
that uniquely covers elements with total profit at least k. The time taken
by the algorithm is O(2k · mk2 ). We thus obtain:
Theorem 5.14. Let (U , F, B, k) be an instance of the Budgeted Unique
Coverage problem with either integral costs and rational profits ≥ 1 or with
rational costs ≥ 1 and integral profits. Then one can find a subfamily F ′
of total cost at most B that uniquely covers elements with total profit at
least k, if there exists one, in time O(min{2O(k log B) , 2O(k log k+k log log B) } ·
Bmk 2 log2 n).
5.6
Algorithms for Special Cases
We now consider deterministic algorithms for two special cases of Budgeted Unique Coverage: Unique Coverage and Budgeted Max
Cut.
5.6.1
Unique Coverage
An instance (U , F, k) of Unique Coverage can be viewed as an instance of
Budgeted Unique Coverage where the costs and profits are all one and
the budget B = k as we do not need more than k sets to cover k elements
uniquely. Using Theorem 5.12, we immediately obtain an algorithm with
running time
O(2O(k log k) · |F| · k2 log2 n).
In this subsection we present an algorithm for Unique Coverage that runs
in deterministic time
O(2O(k log log k) · |F| · k + |F|2 ).
We first need some lower bounds on the number of elements that can be
uniquely covered in any instance of Unique Coverage.
Define the frequency fu of an element u ∈ U to be the number of sets in
the family F that contain u. Let γ denote the maximum frequency, that is,
γ = maxu∈U {fu }.
Lemma 5.15. Given an instance (U , F, k) of Unique Coverage, one
can in polynomial time obtain a subfamily F ′ ⊆ F such that F ′ covers at
least |U |/(4e log γ) elements uniquely, where γ = maxu∈U {fu }.
Proof. The proof of lemma is similar to the Lemma 5.8 but we provide a
proof here for the sake of completeness.
Partition the elements of U into sets C0 , C1 . . . Cr−1 , such that u ∈ Ci
if 2i ≤ fu ≤ 2i+1 − 1. Note that i ranges from 0 to log(γ + 1) − 1, since
the frequency of any element in Cr−1 is at most 2r − 1, which in turn is at
100
Chapter 5. The Unique Coverage Problem
most γ. So the total number of sets that the universe gets partitioned into
is log(γ + 1).
Clearly, there exists j such that |Cj | ≥ n/ log(γ + 1). Fix j to be the
index for which |Cj | ≥ n/ log(γ + 1). Let u ∈ Cj and note that 2j ≤ fu ≤
2j+1 − 1. Construct a subfamily F ′ from F by going through each set in F
and including it in F ′ with probability 1/2j+1 . Denote by lu the probability
that u is covered uniquely by F ′ . Then
(fu −1)
fu
1
1 − 2j+1
lu =
j+1
2
(fu −1)
1
(since fu ≥ 2j ).
≥ 21 1 − 2j+1
(2j+1 −2)
1
≥ 21 1 − 2j+1
(since fu ≤ 2j+1 − 1).
(2j+1 )
1
1
. (by Lemma 5.7).
≥ 12 1 − 2j+1
≥ 4e
Now, let Xu be an indicator random variable which takes value 1 if u is
covered
uniquely by the subfamily F ′ , and 0 otherwise. Also, define X =
P
u Xu .
E(X) =
X
u∈U
E(Xu ) ≥
X
u∈Cj
E(Xu ) =
X
u∈Cj
lu ≥
n
.
4e log γ
This immediately implies the existence of a subfamily that covers n/4e log γ
elements (or more) uniquely. We can derandomize the above algorithm using
the method of conditional probabilities to obtain the desired subfamily.
Lemma 5.16. Given an instance (U , F, k) of Unique Coverage, one
can in polynomial time obtain a subfamily F ′ ⊆ F such that F ′ covers at
least |U |/(8e log M ) elements uniquely, where M = maxS∈F {|S|}.
Proof. We begin by constructing a subfamily F ′ from F that is minimal
in the sense that every set in F ′ covers at least one element in U uniquely.
Such a subfamily is easily obtained, by going over every set in the family and
checking if it has at least one element which is not containedSin any other
set. Let m′ denote the size of the subfamily F ′ . Note that | S∈F ′ S| = n.
For the proof of the lemma we distinguish two cases based on m′ :
Case 1: (m′ ≥ n/2) As the subfamily is minimal, by construction, we are
immediately able to cover at least n/2 elements uniquely. Thus F ′ itself
satisfies the claim of the lemma.
Case 2: (m′ < n/2) In this case, we first claim that |{u ∈ F ′ : fu < M }| ≥
n/2. If not, then there would be more
P than n/2 elements whose frequency
is
Pat least M , which implies that S∈F ′ |S| > M n/2. On the other hand,
S∈F ′ |S| is clearly at most M (n/2 − 1) (because there are strictly less
than n/2 sets in the family and the size of any set in the family is bounded
5.6. Algorithms for Special Cases
101
by M ). The claim implies that there exists a set of at least n/2 elements
whose frequency is less than M . Denote this set of elements by V. Consider
the family F ′′ obtained from F ′ as follows: F ′′ = {S ∩V | S ∈ F ′ }. Applying
Lemma 5.15 to the instance (V, F ′′ ), we obtain a subfamily T of F ′′ that
covers at least n/(8e log M ) elements uniquely. The corresponding subfamily of F ′ will clearly cover the same set of elements uniquely in U . This
completes the proof of the lemma.
Using these lower bounds on the number of elements that are uniquely
covered, we can upper bound the size of a yes-instance of the Unique
Coverage problem as a function of the parameter k. Let (U , F, k) be an
instance of Unique Coverage reduced w.r.t. Rules R1 and R2 described in
Section 5.4. Then |S| ≤ k−1 for all S ∈ F and M is bounded above by k−1.
If k ≤ n/8e log(k − 1), then there exists a subfamily that covers k elements
uniquely. If not, we have k > n/8e log k, which implies that n < 8ek log k.
Lemma 5.17. Let (U , F, k) be an instance of Unique Coverage. Then,
in polynomial time, one can either find a subfamily covering at least k elements uniquely, or an equivalent instance where the size of the universe
is O(k log k).
Note that the above lemma shows that Unique Coverage admits a
kernel with kO(k) sets which is what was shown in Corollary 5.2.
An improved algorithm for Unique Coverage first applies Lemma 5.17
and obtains an instance of Unique Coverage, (U , F, k), where n = |U | ≤
O(k log k). Now we examine all k-sized subsets X of the universe U and
check whether there exists a subfamily that covers it uniquely. Let X =
{ui1 , ui2 , . . . , uik }, and let h be a function that maps X injectively to [k]
and each element in U \ X to the color k + 1. Applying Lemma 5.10 to
the instance (U , F, B = k, k), with the coloring function h described above
gives us an algorithm to find the desired F ′ in time O(2k k2 m). Note that a
factor of k can be avoided by directly applying dynamic programming over
subsets of X. The size of U is upper bounded by 8ek log k and hence
the
total number of subsets that need to be examined is at most 8ek klog k , which
is bounded above by (8e log k)k ≤ 24.5k+k log log k . Combining this with the
above discussion results in:
Theorem 5.15. Let (U , F, k) be an instance of Unique Coverage. Then
one can check whether there exists a subfamily covering at least k elements
of U uniquely in time O(25.5k+k log log k · mk + m2 ).
5.6.2
Budgeted Max Cut
An instance of Budgeted Max Cut consists of an undirected graph G =
(V, E) on n vertices and m edges; a cost function c : V → Z+ ; a profit
function p : E → Z+ ; and positive integers k and B. The question is whether
102
Chapter 5. The Unique Coverage Problem
there exists a cut (T, V \ T ), ∅ =
6 T 6= V , such that the total cost of the
vertices in T is at most B and the total profit of the edges crossing the
cut is at least k. This problem can be modelled as an instance Budgeted
Unique Coverage by taking U = E and F = {Sv : v ∈ V }, where Sv =
{e ∈ E : e is incident on v}.
The algorithm we describe has running time O(2O(k) · Bmk · log2 n).
Given S ⊆ V , we let c(S) denote the total cost of the elements of S. If (S, V \
S) is a cut in a graph G, then p(S, V \ S) is the total profit of edges across
the cut. Define the profit p̂(v) of a vertex v to be the sum of the profits of
all the edges incident on it. We assume that p̂(v) ≤ k − 1 for all v ∈ V , for
otherwise the given instance is a yes-instance trivially.
Lemma 5.18. If (G, B, k, c, p) is a yes-instance of Budgeted Max Cut
then there
exists a cut (S, S \ V ) such that c(S) ≤ B, p(S, V \ S) ≥ k,
S
and | v∈S Sv | ≤ 4k.
Proof. Since we are given a yes-instance of the problem, there exists a
cut (T, T ′ ) such that c(T ) ≤ B and p(T, T ′ ) ≥ k. Call a vertex v of T
redundant if p(T \ v, T ′ ∪ v) ≥ k. Starting with the cut (T, T ′ ), transfer
redundant vertices from T to T ′ and obtain a cut (S, S ′ ) such that S ⊂ T
and S does not contain any redundant vertices. Observe that c(S) ≤ B
and p(S, S ′ ) ≥ k. For any v ∈ S, p̂(v) ≤ k − 1 and p(S \ v, S ′ ∪ v) ≤ k − 1.
Therefore p(S, S ′ ) ≤ 2k. For v ∈ S, partition Sv as Iv ⊎ Cv , where Iv is the
set of edges incident on v that lie entirely in S and Cv are the edges that
lie across the cut (S, S ′ ). Clearly p(Iv ) ≤ p(Cv ), for otherwise, p(S \ v, S ′ ∪
′ ), a contradiction to the fact that S has no redundant vertices.
v) > p(S, SP
P
P
Therefore v∈S p(Iv ) ≤
v∈S p̂(v) ≤ 4k.
v∈S p(Cv ) ≤ 2k.S This yields
Since the profits are at least one, we have | v∈S Sv | ≤ 4k.
We now use the deterministic algorithm outlined before Theorem 5.12
with t = s = 4k and a 4k-perfect hash family by Chen et al. [32]. The
k
running time then works out to O(6.44k log2 n· 4k
k ·2 Bmk) which simplifies
to O(213.8k · Bmk · log2 n).
Theorem 5.16. Let (G, B, k, c, p) be an instance of Budgeted Max Cut.
Then one can find a cut (S, S ′ ) such that c(S) ≤ B and p(S, S ′ ) ≥ k, if there
exists one, in time O(213.8k · Bmk · log2 n).
5.7
Conclusions
In this chapter we studied the parameterized complexity of the Unique
Coverage problem. We considered several plausible parameterizations of
the problem and showed that except for the standard parameterized version (and a generalization of it), all other versions are unlikely to be fixedparameter intractable. We also described problem kernels for several special
5.7. Conclusions
103
Unique Coverage (Parameter: k)
Kernel Size
Sect.
Each element occurs in at most b sets
Intersection size bounded by c
General case
Each set of size at most b
(k − 1)b
k c+1
4k
2b+k
5.4
5.4.1
5.4.2
5.4.2
Budgeted Unique Coverage
Complexity
Sect.
Arbitrary weights (parameters B and k)
Integer weights (parameter B)
Integer weights (parameters B and k)
Not FPT (unless P = NP) 5.5.1
W[1]-hard
5.5.1
FPT
5.5.2
Table 5.1: Main results in this chapter.
cases of the standard parameterized version of Unique Coverage and
showed that the general version admits a kernel with at most 4k sets. We
noted that this is essentially the best possible kernel due to a lower bound
result by Dom et al. [46].
We gave fixed-parameter tractable algorithms for Budgeted Unique
Coverage and several of its variants. Our algorithms were based on an
application of the well-known method of color-coding in an interesting way.
Our randomized algorithms have good running times but the deterministic
algorithms make use of either perfect hash families or (k, s)-hash families and
this introduces large constants in the running times, a common enough phenomenon when derandomizing randomized algorithms using such function
families [6]. The main results of this chapter are summarized in Table 5.1.
Our use of (k, s)-hash families to derandomize algorithms is perhaps the
first application of these hash families outside the domain of coding theory and it suggests that this artificial-looking class of hash families may be
important and may have other uses as well. It will be interesting to give
explicit constructions of (k, s)-hash families of size promised by Lemma 5.12
and explore other applications of our generalization of the color-coding technique.
104
Chapter 5. The Unique Coverage Problem
Chapter 6
The Induced Subgraph Problem in
Directed Graphs
In this chapter and the next, we consider the parameterized complexity of
NP-optimization problems in directed graphs. There have been relatively
few results on parameterized problems on directed graphs since, in general,
many problems that can be formulated for both directed and undirected
graphs are significantly more difficult for directed graphs [78]. For instance,
the Feedback Edge Set problem is polynomial-time solvable in undirected
graphs but NP-complete in directed graphs [64]. From the parameterized
complexity point of view, the Undirected Feedback Vertex Set problem is known to be fixed-parameter tractable but the Directed Feedback
Vertex (Edge) Set was open for a number of years until it was shown to
be in FPT by Chen et al. [31]. The results in this chapter and the next add
to the growing literature on parameterized complexity results on directed
graphs.
The problem that we consider in this chapter actually represents a class
of problems that is loosely termed as the Induced Subgraph problem and
is defined as follows: Given a graph G and a nonnegative integer k, does G
have a vertex induced subgraph of size k satisfying some prespecified property? Lewis and Yannakakis [93] proved that this problem is NP-complete
when the property is nontrivial and hereditary. Khot and Raman [86] studied the parameterized complexity of this problem in undirected graphs and
completely characterized properties for which the problem is in FPT and for
which the problem is W[1]-complete. We extend their result for hereditary
properties on directed graphs. As a corollary of our results, we show, for
example, that the problem of deciding whether an input digraph D has a
transitive induced subdigraph of size k is fixed parameter tractable while the
problem of deciding whether D has a planar induced subdigraph of size k is
W[1]-complete.
This chapter is organized as follows. In Section 6.1, we define the problem
formally, briefly survey some previous work and define the digraph classes
105
106
Chapter 6. The Induced Subgraph Problem
that appear in this chapter. In Section 6.2, we give a complete specification
of when the Induced Subgraph problem is fixed-parameter tractable and
when it is not, for hereditary properties on general directed graphs. In
Section 6.3, we consider the problem for hereditary properties on oriented
graphs. In Section 6.4, we characterize those hereditary properties for which
the Induced Subgraph problem is hard on general digraphs but in FPT
on oriented graphs. We end with some concluding remarks in Section 6.5.
6.1
Problem Definition and Previous Work
A graph property P is an isomorphism-closed set of graphs. A graph property P is nontrivial if there exists an infinite family of graphs satisfying P and
an infinite family not satisfying P. A graph property P is hereditary if G ∈ P
implies that every induced subgraph of G is also in P (see [93]). Examples
of hereditary properties (for undirected graphs) include the class of planar, outerplanar, bipartite, interval, comparability, acyclic, bounded-degree,
chordal, complete, independent set and line invertible graphs [93]. Similarly
for digraphs, the following graph classes are hereditary: acyclic, transitive,
symmetric, anti-symmetric, line-digraph, maximum outdegree r, maximum
indegree r, without cycles of length l, without cycles of length ≤ l [93].
A property P has a forbidden set characterization if there exists a set F
of graphs such that G has property P if and only if no element of F is
an induced subgraph of G. The set F is called the forbidden set of P.
It is well-known that a property P is hereditary if and only if it has a
forbidden set characterization [25]. For if a property P has a forbidden set
characterization, it is clearly hereditary. Conversely suppose P is hereditary
and consider the set S of graphs not in P. The induced subgraph relation
defines a partial order among the elements of S and the minimal elements
of this partial order form the forbidden set of P.
For a property P on (directed) graphs, the Induced Subgraph problem is defined as follows: Given a (directed) graph G find a vertex subset
of maximum size that induces a subgraph with property P. Lewis and Yannakakis [93] proved this problem to be NP-hard when the property P is
nontrivial and hereditary. If, in addition, the given property can be tested
in polynomial time, their results show that the Induced Subgraph problem is NP-complete. The parameterized version of this problem for a given
property P is defined as follows.
P (G, k, P)
Input:
A graph G = (V, E) and a nonnegative integer k ≤ |V |.
Parameter: The integer k.
Question: Does G have an induced subgraph on at least k vertices with
property P?
6.2. General Directed Graphs
107
Call the directed graphs version of this problem P (D, k, P).
Khot and Raman [86] resolved the problem P (G, k, P) when P is a nontrivial hereditary property on undirected graphs. They show that if the
property P either contains all trivial graphs and all cliques or excludes a
trivial graph and a clique then the problem P (G, k, P) is fixed-parameter
tractable and W[1]-complete otherwise. The proof techniques employed by
them make heavy use of Ramsey theory. In particular, they make use of the
fact that any “sufficiently large” undirected graph either contains a trivial
graph or a clique.
In this chapter we consider the problem P (D, k, P) when P is a nontrivial
hereditary property on directed graphs. We give a complete specification of
when the problem P (D, k, P) is fixed-parameter tractable and when it is not.
At this point, we define a few graph classes that we will encounter in this
chapter (see [12] for details). A graph is trivial if it has no edges (or arcs).
A graph without cycles is acyclic. A directed graph D = (V, A) is complete
symmetric if for all distinct u, v ∈ V , we have (u, v) ∈ A and (v, u) ∈ A. A
directed graph is a tournament if for all vertex pairs, there exists exactly
one arc between them. A directed graph with at most one arc between
each vertex-pair is called an oriented graph. That is, an oriented graph
is one which does not have cycles of length two. Oriented graphs are also
called antisymmetric. A digraph D = (V, A) is symmetric if (u, v) ∈ A
implies (v, u) ∈ A.
A set K of vertices in a digraph D = (V, A) is a kernel if K is independent and the closed neighbourhood N [K] of K is V itself. A digraph is
kernel-perfect if every induced subdigraph has a kernel. Clearly all trivial
and complete symmetric digraphs are kernel-perfect, as any maximal independent set is a kernel. What is interesting is that any acyclic digraph is
kernel perfect too [12]. A digraph is chordal if its underlying undirected
graph is chordal, that is, if it has no induced cycles of length greater than
three. A digraph D = (V, A) is transitive if for distinct vertices u, v, w ∈ V ,
(u, v) ∈ A and (v, w) ∈ A imply that (u, w) ∈ A. A digraph is quasitransitive if for all distinct vertices u, v, w, if (u, v) ∈ A and (v, w) ∈ A then
there is at least one arc between u and w.
6.2
General Directed Graphs
We begin with by examining a specialization of Ramsey’s theorem applicable
to directed graphs.
Fact 6.1 ([69]). Suppose that the 2-element subsets of every set S with n elements are partitioned into m disjoint families F1 , . . . , Fm . Let p1 , . . . , pm
be positive integers with pi ≥ 2, 1 ≤ i ≤ m. Then there is a number r(p1 , . . . , pm ), such that for every set S with n ≥ r(p1 , . . . , pm ) elements
108
Chapter 6. The Induced Subgraph Problem
there exists an i, 1 ≤ i ≤ m, and a subset Ai of S with pi elements all of
whose 2-subsets are in the family Fi .
If D = (V, A) is a digraph with V = {u1 , . . . , un }, partition the 2-subsets
of V into four classes, as follows:
F1 = {{ui , uj : (ui , uj ), (uj , ui ) ∈
/ A}}
F2 = {{ui , uj : (ui , uj ), (uj , ui ) ∈ A}}
F3 = {{ui , uj : (ui , uj ) ∈ A, (uj , ui ) ∈
/ A, i < j}}
F4 = {{ui , uj : (ui , uj ) ∈
/ A, (uj , ui ) ∈ A, i < j}}
From Ramsey’s theorem we have
Corollary 6.1. Let p1 , p2 , p3 be positive natural numbers ≥ 2. Then there
exists a positive number r(p1 , p2 , p3 ) such that any directed graph D on at
least r(p1 , p2 , p3 ) vertices contains either a trivial graph of size p1 , or a
complete symmetric digraph of size p2 , or an acyclic tournament of size p3 .
Thus if P is a nontrivial hereditary property on digraphs, then it must
contain either all trivial graphs, all complete symmetric digraphs and all
acyclic tournaments or exactly two of these graph types of all sizes or exactly
one of these graph types of all sizes.
Theorem 6.1. If P is a hereditary property on digraphs that either contains all trivial graphs, all complete symmetric digraphs and all acyclic
tournaments or excludes a graph of each of these three types, then the problem P (D, k, P) is fixed-parameter tractable.
Proof. Suppose P excludes a trivial graph of size c1 , a complete symmetric
digraph of size c2 and an acyclic tournament of size c3 . Then P cannot
contain any digraph D such that |V (D)| ≥ r(c1 , c2 , c3 ) and is therefore
finite. The problem P (D, k, P) can then be decided in constant time.
Therefore assume that P contains all trivial graphs, all complete symmetric digraphs and all acyclic tournaments. If |V (D)| ≥ r(k, k, k)1 then,
by Ramsey’s theorem, the digraph D has either a trivial graph, or a complete symmetric digraph or an acyclic tournament of size k as an induced
subgraph. Thus the given instance is a yes-instance. Otherwise, |V (D)| <
r(k, k, k) and we check all subsets S ⊆ V (D) of size k to see whether D[S]
has property P. This takes time r(k,k,k)
·f (k), where f (k) is the time taken
k
to decide whether a digraph on k vertices has property P. This proves that
the problem P (D, k, P) is fixed-parameter tractable.
1
We do not need to know the number r(k, k, k) exactly. An upper bound on r(k, k, k)
will serve our purpose. A crude upper-bound for the Ramsey number r(k, . . . , k) is ssk .
| {z }
s times
6.2. General Directed Graphs
109
Corollary 6.2. Given any directed graph D and an integer k, it is fixedparameter tractable to decide whether D has an induced subdigraph on k
vertices that is (1) a kernel perfect digraph, (2) a chordal digraph, (3) a
transitive digraph, or (4) a quasi-transitive digraph.
6.2.1
W[1]-Completeness Results
We show that if the property P contains exactly two of the graph types
of all sizes or exactly one of the graph types of all sizes then the problem P (D, k, P) is W[1]-complete. To do this, we first show that the problem P (D, k, P) is in W[1] for any nontrivial decidable hereditary property P.
We next show that the problem is W[1]-hard by exhibiting a parametric reduction from a W[1]-hard problem.
Lemma 6.1. Let P be a nontrivial decidable hereditary property on digraphs. Then the problem P (D, k, P) is in W[1].
Proof. We reduce P (D, k, P) to the Short Turing Machine Acceptance problem (defined below) which is complete for the class W[1] [47]
(also see Chapter 1).
Input:
A nondeterministic Turing machine M , a string x and nonnegative integer k
Parameter: The integer k.
Question:
Does M have a computation path accepting x in at most k
steps?
Let (D = (V, E), k) be an instance of P (D, k, P), with |V | = n. We will show
that we can construct an instance (MD , x, k′ ) of Short Turing Machine
Acceptance in time O(f (k) · nO(1) ) such that D has an induced subgraph
of size k satisfying property P if and only if MD accepts x within k′ steps,
where k′ depends only on k.
First note that since we assumed P to be decidable, there exists a deterministic Turing machine (DTM) M ′ that takes a digraph D as input and
in time t(|V (D)|) decides whether D satisfies P. The input alphabet of MD
consists of the n + 1 symbols 1, 2, 3, . . . , n, ♯. The NTM MD performs the
following steps.
1. MD nondeterministically writes a sequence of k numbers on its tape
out of its tape alphabet {1, 2, . . . , n}.
2. It then verifies whether the k numbers it has picked are distinct.
3. It then constructs the subgraph D ′ of D represented by these k vertices.
110
Chapter 6. The Induced Subgraph Problem
4. MD passes control to M ′ which then verifies whether D ′ satisfies P.
If yes, MD accepts.
The time taken in Steps 1, 2 and 4 are, respectively, O(k), O(k2 ) and t(k).
Assuming that the graph D is hardwired in MD as an adjacency matrix,
Step 3 takes time O(k2 ).
It is easy to see that (D, k) is a yes-instance of the problem P (D, k, P)
if and only if the NTM MD accepts the empty string in k′ = O(k + k2 + t(k))
steps.
To prove W[1]-hardness, we consider the following four cases: the property P contains
1. all complete symmetric digraphs but not all trivial graphs;
2. all trivial graphs but not all complete symmetric digraphs;
3. all acyclic tournaments but not all trivial graphs;
4. all trivial graphs but not all acyclic tournaments.
Note that cases 1 through 4, though not mutually exclusive, are exhaustive.
We first show that P (D, k, P) is W[1]-hard in cases 1 and 2.
Theorem 6.2. Let P be a hereditary property on digraphs that contains
all complete symmetric digraphs but not all trivial graphs or vice versa.
Then P (D, k, P) is W[1]-complete.
Proof. Membership in W[1] was shown in Lemma 6.1. We therefore need
only establish W[1]-hardness.
Let P be a property on digraphs. Define P1 as follows. An undirected
graph G ∈ P1 if and only if the directed graph D obtained from G by
replacing every edge {u, v} ∈ E(G) by the arcs (u, v) and (v, u) is in P.
Note that G contains a clique of size k if and only if D contains a complete
symmetric digraph of size k and G contains an independent set of size k if
and only if D contains an independent set of size k. Also note that
1. P1 is nontrivial and hereditary if and only if P is nontrivial and hereditary,
2. P1 contains all cliques but not all trivial graphs if and only if P contains
all complete symmetric digraphs but not all trivial graphs, and
3. P1 contains all trivial graphs but not all cliques if and only if P contains
all trivial graphs but not all complete symmetric digraphs.
6.2. General Directed Graphs
111
By Khot and Raman [86], the problem P1 (G, k, P1 ) is W[1]-hard when P1
contains all cliques but not all trivial graphs or vice versa.
We now exhibit a parametric reduction from P1 (G, k, P1 ) to P (D, k, P).
Let (G, k) be an instance of P1 . Construct a directed graph D as follows:
V (D) = V (G) and for all u, v ∈ V (G), if {u, v} ∈ E(G) add the arcs (u, v)
and (v, u) in A(D). The directed graph D has no other arcs. From the
manner in which property P1 was defined, it is clear that G has an induced subgraph on k vertices satisfying P1 if and only if D has an induced
subdigraph on k vertices satisfying P. This completes the proof.
We next show that P (D, k, P) is W[1]-hard in cases 3 and 4.
Theorem 6.3. Let P be a hereditary property on digraphs that contains
all acyclic tournaments but not all trivial graphs or vice versa. Then the
problem P (D, k, P) is W[1]-complete.
Proof. As before, we show only W[1]-hardness. Let P be a property on
digraphs. Define P1 to be a set of undirected graphs with the following
property: An undirected graph G ∈ P1 if and only if the directed graph D ∈
P, where V (D) = V (G) with an ordering on the vertices and
A(D) = {(u, v) : u < v and {u, v} ∈ E(G)}.
Clearly G has an independent set of size k if and only if D has an independent
set of size k and G has a clique of size k if and only if D has an acyclic
tournament of size k. Also
1. P1 is nontrivial and hereditary if and only if P is nontrivial and hereditary,
2. P1 contains all trivial graphs but not all cliques if and only if P contains
all trivial graphs but not all acyclic tournaments, and
3. P1 contains all cliques but not all trivial graphs if and only if P contains
all acyclic tournaments but not all trivial graphs.
The problem P1 (G, k, P1 ) is W[1]-hard by [86] when P1 contains all trivial
graphs but not all cliques or vice versa.
We now exhibit a parametric reduction from the problem P1 (G, k, P1 )
to the problem P (D, k, P). Given an instance (G, k) of P1 (G, k, P1 ), let D
be the directed graph obtained by orienting the edges of G from lower ordered vertices to higher ordered vertices. From the manner in which we
constructed P1 , it is easy to see that G has an induced subgraph on k vertices satisfying P1 if and only if D has an induced subdigraph on k vertices
satisfying P. This proves the theorem.
112
Chapter 6. The Induced Subgraph Problem
We now look at some applications. The set of symmetric digraphs
contains all trivial graphs and all complete symmetric digraphs but no
acyclic tournament. The following hereditary properties contain all trivial
graphs and acyclic tournaments but not all complete symmetric digraphs:
(1) acyclic digraphs, (2) oriented digraphs, (3) digraphs without dicycles of
length l and (4) digraphs without dicycles of length ≤ l. Hence the following
corollary is immediate from Theorems 6.2 and 6.3.
Corollary 6.3. Given a digraph D and a positive integer k, it is W[1]complete to decide whether D has an induced subdigraph of size k that is
(1) a symmetric digraph, (2) acyclic, (3) an oriented digraph, (4) without
dicycles of length l, or (5) without dicycles of length ≤ l.
The following digraph properties contain all trivial graphs but not all
complete symmetric digraphs and acyclic tournaments: (1) with maximum
indegree r, (2) with maximum outdegree r, (3) bipartite, (4) colorable with c
colors, for some constant c ≥ 1, (5) planar, (6) a line digraph. Hence the
following corollary is immediate from Theorem 6.3.
Corollary 6.4. Given a digraph D and a positive integer k, it is W[1]complete to decide whether D has an induced subdigraph of size k that is
(1) of maximum indegree r, (2) of maximum outdegree r, (3) bipartite,
(4) colorable with c colors, for some constant c ≥ 1, (5) planar, or (6) a line
digraph.
6.3
Oriented Graphs
Though Corollary 6.3 says that finding an acyclic subdigraph of size at
least k is hard in general digraphs, Raman and Saurabh [114] have shown
that this problem is in FPT in oriented graphs. In this section, we look at
the general Induced Subgraph problem in oriented graphs.
Recall that an oriented graph is a directed graph in which every pair
of vertices has at most one arc between them. Thus oriented graphs are
precisely those digraphs with no 2-cycle. For oriented graphs, Ramsey’s
theorem says: For positive integers p and q there exists an integer r(p, q) ∈ N
such that any oriented graph on at least r(p, q) vertices either has a trivial
graph of size p or an acyclic tournament of size q.
Any nontrivial hereditary property P on oriented graphs can therefore
be classified into one of the three types: (1) P contains all trivial graphs
and all acyclic tournaments; (2) P contains all trivial graphs but not all
acyclic tournaments; (3) P contains all acyclic tournaments but not all trivial graphs. As one might suspect, the problem P (D, k, P) is fixed-parameter
tractable for Case (1) and W[1]-complete for Cases (2) and (3). Membership
in W[1] can be easily proved by a parametric reduction to the Short Turing Machine Acceptance Problem similar to the proof of Lemma 6.1.
6.4. General Digraphs vs Oriented Graphs
113
Theorem 6.4. Let P be a hereditary property on oriented graphs that either
contains all trivial graphs and all acyclic tournaments or excludes a trivial graph and an acyclic tournament. Then P (D, k, P) is fixed-parameter
tractable.
Proof. Suppose P excludes a trivial graph of size c1 and an acyclic tournament of size c2 . Then P cannot contain any oriented graph D such
that |V (D)| ≥ r(c1 , c2 ) and is therefore finite. The problem P (D, k, P)
can then be decided in constant time.
Therefore assume that P contains all trivial graphs and all acyclic tournaments. If |V (D)| ≥ r(k, k) then, by Ramsey’s theorem, D has either a
trivial graph or an acyclic tournament of size k as an induced subgraph.
Thus the given instance is a yes-instance. Otherwise, |V (D)| < r(k, k) and
we check all subsets S ⊆ V (D) of size k to see whether D[S] has prop
erty P. This takes time r(k,k)
· f (k), where f (k) is the time taken to decide
k
whether an oriented graph on k vertices has property P. This proves that
the problem P (D, k, P) is fixed-parameter tractable.
Theorem 6.5. Let P be a hereditary property on oriented graphs that contains all trivial graphs but not all acyclic tournaments or vice versa. Then
the problem P (D, k, P) is W[1]-complete.
Proof. Let P be a property on oriented graphs. Define a property P ′ on
undirected graphs as follows: An undirected graph G satisfies P ′ if and
only if the directed graph D satisfies P, where V (D) = V (G) and A(D) =
{(u, v) : u < v, {u, v} ∈ E(G)}. Clearly G has an independent set of size k if
and only if D has an independent set of size k and G has a clique of size k
if and only if D has an acyclic tournament of size k. Also
1. P ′ is nontrivial and hereditary if and only if P is nontrivial and hereditary,
2. P ′ contains all trivial graphs but not all cliques if and only if P contains
all trivial graphs but not all acyclic tournaments, and
3. P ′ contains all cliques but not all trivial graphs if and only if P contains
all acyclic tournaments but not all trivial graphs.
The problem P (G, k, P ′ ) is W[1]-hard by Khot and Raman [86] and it is
easy to see that P (G, k, P ′ ) ≤FPT P (D, k, P).
6.4
General Digraphs vs Oriented Graphs
In this section, we characterize general digraph properties for which the
problem P (D, k, P), when restricted to oriented graphs, becomes fixedparameter tractable. In what follows, if P is a property on general directed
114
Chapter 6. The Induced Subgraph Problem
graphs then its restriction P ′ to oriented graphs is defined to be the set of
all oriented graphs satisfying P.
Corollary 6.5. Let P be a nontrivial hereditary property on digraphs such
that the induced subgraph problem, P (D, k, P), is W[1]-complete. If P ′ is
its restriction to oriented graphs then the problem P (D, k, P ′ ), restricted to
oriented graphs, is fixed-parameter tractable if and only if either one of the
following conditions is satisfied: (1) P contains all trivial graphs and all
acyclic tournaments but not all complete symmetric digraphs, or (2) P contains all complete symmetric digraphs but not all trivial graphs and acyclic
tournaments.
Proof. (⇐) If P satisfies (1), then P ′ contains all trivial graphs and all
acyclic tournaments. If P satisfies (2), then P ′ is finite. The FPT result
then follows from Theorem 6.4.
(⇒) If P satisfies neither (1) nor (2) of the theorem and, if the problem P (D, k, P) is W[1]-complete, then P ′ is a nontrivial hereditary property
on oriented graphs that satisfies either all trivial graphs but not all acyclic
tournaments or vice versa. The hardness proof then follows from Theorem 6.5.
Acyclic digraphs form an example of a hereditary property that contains all trivial graphs and acyclic tournaments but no complete symmetric digraphs. Consequently, the Induced Acyclic Subgraph problem is
W[1]-complete on general directed graphs but in FPT on oriented graphs.
6.5
Conclusion
In this chapter we characterized hereditary properties on digraphs for which
finding an induced subdigraph with k vertices in a given digraph is W[1]complete. We first did this for general directed graphs and then for oriented
graphs. We also characterized hereditary properties for which the induced
subgraph problem is W[1]-complete on general directed graphs but in FPT
for oriented graphs.
A related problem is the Graph Modification problem P(i, j, k) which
asks whether a given input graph G can be ‘modified’ by deleting at most i
vertices, j edges and adding at most k edges so that the resulting graph
satisfies property P. More formally, this problem is defined as follows: Given
an undirected graph G = (V, E) does there exist V ′ ⊆ V , E ′ ⊆ E and E ′′ ⊆
E c (the edge set of the complement graph) with |V ′ | ≤ i, |E ′ | ≤ j and |E c | ≤
k, such that G − V ′ − E ′ ∪ E ′′ satisfies P?
Cai [25] has shown that if a hereditary property has a finite forbidden set,
the graph modification problem P(i, j, k) is fixed-parameter tractable with
parameters i, j, k. There is no general result for the case when the forbidden
6.5. Conclusion
115
set is infinite. For instance, the Odd Cycle Transversal problem is
fixed-parameter tractable [117] whereas the Wheel-Free Vertex (Edge)
Deletion problem is W[2]-hard [95]. It would be interesting to obtain a
dichotomy result for the infinite forbidden set case.
The graph modification problem can be framed for directed graphs as
well (for directed graphs, E c can be viewed as the set of all arcs not in input
digraph D). For example, the well-known Directed Feedback Vertex
Set problem can be cast as the problem P(k, 0, 0), where P is the set of all
acyclic digraphs. It would be interesting to investigate the parameterized
complexity of the Graph Modification problem in directed graphs.
116
Chapter 6. The Induced Subgraph Problem
‘
Chapter 7
The Directed Full Degree Spanning
Tree Problem
We continue our study of the parameterized complexity of NP-optimization
problems in directed graphs in this chapter. Here we consider a directed
analog of the Full Degree Spanning Tree problem where, given a digraph D and a non-negative integer k, the goal is to construct a spanning
out-tree T of D such that at least k vertices in T have the same out-degree
as in D. We show that this problem is W[1]-hard on the class of directed
acyclic graphs and strongly connected digraphs. In the dual version, called
Reduced Degree Spanning Tree, we parameterize from the other extreme and here the goal is to construct a spanning out-tree T such that at
most k vertices in T have out-degrees that are different from that in D. We
show that this problem is fixed-parameter tractable and admits a problem
kernel with at most 8k vertices on strongly connected digraphs and O(k2 )
vertices on general digraphs. We also give an algorithm for this problem on
general digraphs with running time O(5.942k · nO(1) ), where n is the number
of vertices in the input digraph.
7.1
Problem Definition and Previous Work
The Full Degree Spanning Tree problem asks, given a connected undirected graph G = (V, E) and a non-negative integer k as inputs, whether G
has a spanning tree T in which at least k vertices have the same degree in T
as in G. This problem was first studied by Pothof and Schut [112] in the
context of water distribution networks where the goal is to determine the
flow in a network by installing a small number of flow-meters. It so happens
that to measure the flow in each pipe of the network, it is sufficient to find a
spanning tree of the network and install flow-meters at those vertices whose
degree in the spanning tree is smaller than that in the network. To find the
optimal number of flow-meters (which is an expensive equipment) one needs
to find a spanning tree with the largest number of vertices of full degree.
117
118
Chapter 7. The Directed Full Degree Spanning Tree Problem
This problem has attracted a lot of attention [16, 23, 88, 72, 65]. Bhatia
et al. [16] studied this problem from the point-of-view of approximation
√
algorithms and gave a factor-Θ( n) algorithm for it, where n in the number
of vertices in the input graph. They also showed that this problem admits
no factor O(n1/2−ǫ ) approximation algorithm unless NP = co-R. For planar
graphs, a polynomial-time approximation scheme (PTAS) was presented.
Independently, Broersma et al. [23] developed a PTAS for planar graphs and
showed that this problem can be solved in polynomial time in special classes
of graphs such as bounded treewidth graphs and co-comparability graphs.
Guo et al. [72] studied the parameterized complexity of this problem and
showed it to be W[1]-hard. Gaspers et al. [65] give an O(1.9465n · nO(1) )
algorithm for the optimization version of this problem.
One can parameterize the d-FDST problem from the “other end” and
ask whether a graph G has spanning tree T in which at most k vertices have
degrees different from that in G. This problem has been studied under the
name Vertex Feedback Edge Set and is defined as follows. Given a
connected undirected graph G = (V, E) and a nonnegative integer k, find an
edge subset E ′ incident on at most k vertices such that G[E \ E ′ ] is acyclic.
Note that if there exists such an edge set E ′ , then there exists E ′′ ⊆ E ′ such
that G[E \ E ′′ ] is a spanning tree. Khuller et al. [88] show that this problem
is MAX SNP-hard and describe a (2 + ǫ)-approximation algorithm for it for
any fixed ǫ > 0. Guo et al. [72] show that this problem is fixed-parameter
tractable by demonstrating a problem kernel with at most 4k vertices.
Here we consider a natural generalization of these problems to directed
graphs. An oriented tree is a tree in the undirected sense each of whose edges
has been assigned a direction. We say that a subdigraph T of a directed
graph D = (V, A) is an out-tree if it is an oriented tree with exactly one
vertex s of in-degree zero (called the root). An out-tree that contains all
vertices of D is an out-branching of D. Given a digraph D = (V, A) and an
out-tree T of D, we say that a vertex v ∈ V is of full degree if its out-degree
in T is the same as that in D; otherwise, v is said to be of reduced degree. We
define the Directed Full Degree Spanning Tree (d-FDST) problem
as follows.
Input:
Given a directed graph D = (V, A) and an integer k.
Parameter: The integer k.
Question:
Does there exist an out-branching of D in which at least k
vertices are of full degree?
We call the dual of this problem the Directed Reduced Degree Spanning Tree (d-RDST) problem.
Input:
Given a directed graph D = (V, A) and an integer k.
Parameter: The integer k.
7.1. Problem Definition and Previous Work
Question:
119
Does there exist an out-branching of D in which at most k
vertices are of reduced degree?
We show that d-FDST is W[1]-hard, by a reduction from the Independent Set problem, for two important digraph classes: directed acyclic
graphs (DAGs) and strongly connected digraphs. We show that d-RDST
is fixed-parameter tractable (FPT) by exhibiting a problem kernel with at
most O(k2 ) vertices. For strongly connected digraphs, d-RDST admits a
kernel with at most 8k vertices. We also develop an algorithm for d-RDST
with running time O(5.942k · nO(1) ).
Related Results. The Full Degree Spanning Tree problem is one of
the many variants of the generic Constrained Spanning Tree problem,
where one is required to find a spanning tree of a given (di)graph subject
to certain constraints. This class of problems has been studied intensely [3,
37, 41, 52, 58, 73, 113].
In [52], the authors consider the problem Max Leaf Spanning Tree
where one is required to find a spanning tree of an undirected graph with the
maximum number of leaves. When parameterized by the solution size, this
problem admits a kernel with 3.75k vertices. In the directed variant of this
problem, one has to decide whether an input digraph D has an out-branching
with at least k leaves. This problem admits a kernel with O(k3 ) vertices,
provided the root of the out-branching is given as part of the input [58],
and has an algorithm with run-time O(3.72k · nO(1) ) [41]. Another such
problem is Max Internal Spanning Tree, where the objective is to find
a spanning tree (or an out-branching, in case of digraphs) with at least k
internal vertices. For undirected graphs, a 3k-vertex kernel and an algorithm
with running time O(8k · nO(1) ) is known for this problem [61]. For directed
graphs, an O(k2 )-vertex kernel due to [73] and an algorithm with running
time O(40k · nO(1) ) due to [37] is known.
Organization of this Chapter. In Section 7.2 we define the relevant notions
related to digraphs. In Section 7.3 we show that d-FDST is W[1]-hard on two
classes of digraphs: directed acyclic graphs and strongly connected digraphs.
In Section 7.4 we show that the d-RDST problem is in FPT by demonstrating
a kernel with at most O(k2 ) vertices. We first demonstrate a kernel with 8k
vertices for strongly connected digraphs and use the ideas therein to develop
the O(k2 ) kernel for general digraphs. In Section 7.5 we develop an algorithm
for the d-RDST problem with running time O(5.942k · nO(1) ). Finally in
Section 7.6, we end with some concluding remarks and open questions.
120
7.2
Chapter 7. The Directed Full Degree Spanning Tree Problem
Digraphs: Basic Terminology
The notation and terminology that we follow are from [12]. Given a digraph D we let V (D) and A(D) denote the vertex set and arc set, respectively, of D. If u, v ∈ V (D), we say that u is an in-neighbour (outneighbour ) of v if (u, v) ∈ A(D) ((v, u) ∈ A(D)). The in-degree d− (u)
(out-degree d+ (u)) of u is the number of in-neighbours (out-neighbours)
of u. Given a subset V ′ ⊆ V (D), we let D[V ′ ] denote the digraph induced
on V ′ . The underlying undirected graph U (D) is the undirected graph obtained from D by disregarding the orientation of arcs and deleting an edge
for each pair of parallel edges in the resulting graph. The connectivity components of D are the subdigraphs induced by the vertices of components
of U (D).
A digraph is oriented if every pair of vertices has at most one arc between
them. A (v1 , vs )-walk in D = (V, A) is a sequence v1 , . . . , vs of vertices such
that (vi , vi+1 ) ∈ A for all 1 ≤ i ≤ s − 1. A dicycle is a walk v1 , v2 , . . . , vs
such that s ≥ 3, the vertices v1 , . . . , vs−1 are distinct and v1 = vs . A digraph
with no dicycles is called a directed acyclic graph (DAG). A digraph D is
strongly connected if for every pair of distinct vertices u, v ∈ V (D), there
exists a (u, v)-walk and a (v, u)-walk. A strong component of a digraph is
a maximal induced subdigraph that is strongly connected. The strong component digraph SC(D) is the directed acyclic graph obtained by contracting
each strong component to a single vertex and deleting any parallel arcs obtained in this process. A strong component S of a digraph D is a source
strong component if no vertex in S has an in-neighbour in V (D) \ V (S).
The following is a necessary and sufficient condition for a digraph to have
an out-branching.
Proposition 7.1 ([12]). A digraph D has an out-branching if and only if D
has a unique source strong component.
One can obtain the strongly connected components of a digraph D in
time O(n + m) [38], where n = |V (D)| and m = |A(D)|. Each strong
component can be stored as an n-bit vector where the ith bit is a one if
and only if vertex i is in the strong component. It is now easy to see that
one can verify in (n + m) time whether there exists a unique source strong
component.
7.3
The d-FDST Problem
We now show that d-FDST is W[1]-hard on two important digraph classes:
DAGs and strongly connected digraphs. This is a modification of the reduction presented in [88] (Lemma 3.2).
7.3. The d-FDST Problem
121
a
x
v1
e1
u
v
e
vn
em
Figure 7.1: The digraph D.
Theorem 7.1. The d-FDST problem parameterized by the solution size is
W[1]-hard on directed acyclic graphs (DAGs) and strongly connected digraphs. Also the d-RDST problem is NP-hard on strongly connected digraphs.
Proof. We show that k-Independent Set, which is known to be W[1]complete [47], fixed-parameter reduces to the d-FDST problem. Let (G, k)
be an instance of the k-Independent Set problem where we assume G to
be a connected undirected graph on n vertices and m edges. Construct a
directed graph D as follows. The vertex set V (D) consists of n + m + 2
vertices: v1 , . . . , vn , e1 , . . . , em , a, x, where the vertices vi , for 1 ≤ i ≤ n,
and ej , for 1 ≤ j ≤ m, “correspond”, respectively, to the vertices and edges
of G and a, x are two special vertices. The digraph D can be viewed as a
three-layer graph. Layer one consists of vertex a. Layer two consists of the
vertices x, v1 , . . . , vn and vertex a has an out-arc to each vertex in layer two.
Layer three consists of the vertices e1 , . . . , em and each ej , for 1 ≤ j ≤ m,
has an out-arc to vertex x. If e = {u, v} ∈ E(G) then the vertices u and v
in layer two have an out-arc each to vertex e in layer three. This completes
the description of D. It is easy to verify that D is a DAG.
Observe that in any out-branching T of D every vertex (except the root)
has exactly one in-neighbor and that:
1. Vertices a, e1 , . . . , em are the only vertices of D of in-degree zero and
therefore the root of T must be one of these. Moreover, if a preserves
its out-degree in T then vertices e1 , . . . , em must be of reduced degree.
2. At most one vertex from among e1 , . . . , em can preserve its out-degree
in T because each of them has an out-arc to x.
3. Vertex x preserves its out-degree in T because x is of out-degree zero.
122
Chapter 7. The Directed Full Degree Spanning Tree Problem
We claim that G has an independent set of size k if and only if the
digraph D has an out-branching with k + 2 vertices of full degree. Suppose
that G has a k-independent set on the vertices vi1 , . . . , vik . Consider the
subdigraph T ′ induced by the vertices a, vi1 , . . . , vik and their out-neighbors.
It is easy to verify that T ′ is actually an out-tree rooted at a in which
vertices a, x, vi1 , . . . , vik have full degree. For each edge e ∈ E(G) that is not
incident to any vertex in the k-independent set, arbitrarily choose one of its
endpoints, say v, and add the arc (v, e) to the out-tree T ′ . This converts
the out-tree T ′ into an out-branching T with at least k + 2 vertices of full
degree. Conversely suppose that D admits an out-branching T in which at
least k + 2 vertices preserve their out-degree. We consider two cases.
Case 1. Vertex a preserves its out-degree. Then no vertex from layer
three preserves its out-degree, as each of these vertices has an out-arc to x.
Since x is the other vertex of full degree, it must be that k vertices from
among v1 , . . . , vn preserve their out-degree. No pair from among these k
vertices form an edge in G, for otherwise, they would have an out-arc to the
same vertex e in layer three in T and this would contradict the assumption
that T is an out-branching. Hence these k vertices must be independent
in G.
Case 2. Vertex a does not preserve its out-degree. By Observation 2, at
most one vertex from layer three can preserve its out-degree and, by Observation 3, x preserves its out-degree in every out-branching. Hence at least k
vertices from among the v1 , . . . , vn preserve their out-degree. These vertices
from a k-independent set in G. This shows that d-FDST is W[1]-hard on
DAGs.
By modifying the above reduction from k-Independent Set, we show
that d-FDST is W[1]-hard on the class of strongly connected digraphs (and
hence that the d-RDST problem is NP-hard on this class of digraphs). Given
an instance (G, k) of k-Independent Set, it is no loss of generality to
assume that G is a connected non-bipartite graph. Construct the digraph D
as above with just one modification: add the arc (x, a). It is easy to verify
that the resulting digraph is strongly connected. Then G has an independent
set of size k if and only if D admits an out-branching with k+2 vertices of full
degree. Suppose G has a k-independent set on the vertex set {vi1 , . . . , vik }.
Since G is non-bipartite, there exists an edge ej both of whose endpoints
are in V (G) \ {vi1 , . . . , vik }. It is easy to see that there is an out-branching
with ej as root in which the vertices ej , x, vi1 , . . . , vik are of full degree.
Conversely suppose that D has an out-branching with k + 2 vertices of full
degree. Between a and x, at most one can preserve its out-degree and
among a, e1 , . . . , em at most one can preserve its out-degree. Therefore at
least k vertices from among v1 , . . . , vn preserve their out-degree. These
vertices form an independent set in G. This shows that d-RDST is NP-hard
on strongly connected digraphs and completes the proof of the theorem.
7.4. d-RDST: A Problem Kernel
7.4
123
d-RDST: A Problem Kernel
In this section we show that d-RDST admits a problem-kernel with O(k2 )
vertices and is therefore fixed-parameter tractable. We first consider the
special case when the input digraph is strongly connected and establish a
kernel with 8k vertices for this case. This will give some insight as to how
to tackle the general case.
Observe that if (D, k) is a yes-instance of the d-RDST problem and T
is a solution out-branching (one in which at most k vertices are of reduced
out-degree), then the subdigraph of D induced by the vertices of full degree
is a forest in the undirected sense and hence has treewidth one (for more on
treewidth, see [20, 91]). Therefore the underlying undirected graph U (D)
has treewidth at most k + 1. Moreover one can show that the property of
having an out-branching with at most k vertices of reduced out-degree is
expressible in monadic second-order logic [39]. One can now use the results
of Arnborg et al. [7] to conclude that for every fixed k the d-RDST problem can be decided in linear time. This shows that the d-RDST problem
is fixed-parameter tractable. However the running time dependence of this
algorithm on k is huge making it impractical. In what follows, we give an
alternative algorithm with a more well-behaved dependence on the parameter k.
7.4.1
A Linear Kernel for Strongly Connected Digraphs
We actually establish the 8k-vertex kernel for a more general class of digraphs, those in which every vertex has out-degree at least one. Call this
class of digraphs out-degree at least one digraphs and denote it by D1+ . It
is easy to see that strongly connected digraphs (SCDs) is a subclass of D1+ .
Since a digraph in D1+ can have vertices of in-degree zero, it follows that
SCDs form a proper subclass of D1+ .
There are three simple reduction rules for the case where the input is an
out-degree at least one digraph. We assume that the input is (D, k).
Rule 1. If there exists u ∈ V (D) such that d− (u) ≥ k + 2 then return no;
else return (D, k).
Rule 2. If there are k + 1 vertices of out-degree at least k + 1 then return
no; else return (D, k).
Rule 3 (The Path Rule). Let x0 , x1 . . . , xp−1 , xp be a sequence of vertices in D such that for 0 ≤ i ≤ p − 1 we have d+ (xi ) = 1 and
(xi , xi+1 ) ∈ A(D). Let Y0 be the set of in-neighbours of x1 , . . . , xp−1
and let Y := Y0 \ {x0 , x1 , . . . , xp−2 }. Delete the vertices x1 , . . . , xp−1
and add two new vertices z1 , z2 and the arcs (x0 , z1 ), (z1 , z2 ), (z2 , xp ).
For y ∈ Y , if y has at least two out-neighbors in {x1 , . . . , xp−1 },
124
Chapter 7. The Directed Full Degree Spanning Tree Problem
x0
x0
x1
z1
x2
y1
y1
y2
y2
z2
xp−1
xp
xp
Figure 7.2: Illustrating the Path Rule: the left and right-hand sides show, respectively, the situation before and after the transformation. Vertex y1 has two
neighbors and vertex y2 just one neighbor in the set {x1 , . . . , xp−1 }.
then add arcs (y, z1 ), (y, z2 ); if y has exactly one out-neighbor in
{x1 , . . . , xp−1 }, then add the arc (y, z1 ). Return (D, k). See Figure 7.2.
It is easy to see that Rules 1 and 2 are indeed reduction rules for the
d-RDST problem on out-degree at least one digraphs. If a vertex v has
in-degree at least k + 2 then at least k + 1 in-neighbors of u must be of
reduced degree in any out-branching. This shows that Rule 1 is a reduction
rule. If a vertex u has out-degree k + 1 and is of full degree in some outbranching T then T has at least k + 1 leaves. Since the input digraph is such
that every vertex has out-degree at least one, this means that in T there
are at least k + 1 vertices of reduced degree. This shows that any vertex
of out-degree k + 1 must necessarily be of reduced degree in any solution
out-branching. Therefore if there are k + 1 such vertices, the given instance
is a no-instance. This proves that Rule 2 is a reduction rule.
Lemma 7.1. Rule 3 is a reduction rule for the d-RDST problem.
Proof. It is sufficient to show that if (D ′ , k) is the instance obtained by one
application of Rule 3 to an instance (D, k), then D has an out-branching
with at most k vertices of reduced out-degree if and only if D ′ has an outbranching with at most k vertices of reduced degree.
Suppose D ′ has an out-branching T ′ with at most k vertices of reduced
degree. There are two cases to consider. In the first case, there are no
arcs from Y to z1 or z2 in T ′ . In this case we may assume without loss
of generality that the path x0 → z1 → z2 occurs as a sub-path of T ′ . For
if x0 → z1 → z2 is not a subpath of T ′ , then one of z1 or z2 has in-degree
7.4. d-RDST: A Problem Kernel
125
zero in T ′ . Hence it must be that either z1 or z2 is the root of T ′ . If z1 is the
root of T ′ then x0 is a leaf in T ′ ; if z2 is the root then z1 is a leaf. In either
case, we can make x0 the root and maintain the path x0 → z1 → z2 without
increasing the number of vertices of reduced degree. In order to construct
an out-branching T for D, replace x0 → z1 → z2 by the path x0 → x1 →
· · · → xp−1 . Moreover if T ′ contains the arc (z2 , xp ) then, in constructing T ,
add the arc (xp−1 , xp ). Note that T has at most k vertices of reduced degree.
In the second case, there exists at least one vertex y ∈ Y with arcs
to {z1 , z2 }. Suppose that T ′ contains the arcs (y1 , z1 ), (y2 , z2 ), where y1
and y2 are (not necessarily distinct) vertices in Y . Note that both x0
and z1 are of out-degree zero in T ′ and hence of reduced degree. Observe
that T ′ \ {z1 } is an out-branching for D ′ \ {z1 } as z1 is a leaf in T ′ . We
transform T ′ into another out-branching for D ′ by deleting the arc (y1 , z1 )
and adding the arc (x0 , z1 ). In this new out-branching, x0 is of full degree
and y1 is of reduced degree but the number of vertices of reduced degree
does not increase.
We can therefore assume without loss of generality that in T ′ there is
exactly one vertex y ∈ Y with an out-arc to {z1 , z2 }. Suppose (y, z2 ) ∈
A(T ′ ). Then y must be of reduced degree as whenever we have an arc (y, z2 ),
we also have an arc (y, z1 ). In this case we transform T ′ by deleting the
arcs (y, z2 ), (x0 , z1 ) and introducing the arcs (y, z1 ), (z1 , z2 ). The resulting
digraph is an out-branching with at most k vertices of reduced degree as
x0 now is of reduced degree but z1 is of full degree. Therefore we are left
to consider the case when y has an arc to z1 only. Let xs be the first
out-neighbor of y in {x1 , . . . , xp−1 }. Delete z1 , z2 and connect x0 to the
dipath x1 → · · · → xs−1 and y to the dipath xs → · · · → xp−1 . Add the
arc (xp−1 , xp ) if (z2 , xp ) ∈ A(T ′ ). The resulting digraph is an out-branching
for D with at most k vertices of reduced degree.
To prove the converse, suppose that D has an out-branching T with at
most k vertices of reduced degree. Again there are two cases to consider.
Case 1. There are no arcs from Y to any xi , for 1 ≤ i ≤ p−1, in T . There are
two sub-cases here. Either T contains the dipath x0 → x1 → · · · xp−1 → xp ,
in which case we can compress it to the path (x0 , z1 , z2 , xp ) to obtain an outbranching T ′ for D ′ with at most k vertices of reduced degree. Otherwise
one of the vertices x1 , . . . , xp must be the root of T . If xp is the root, then T
contains the dipath x0 → x1 → · · · xp−1 and we replace it by (x0 , z1 , z2 ) to
obtain an out-branching T ′ of D ′ . If one of x1 , . . . , xp−1 is the root, then
delete x1 , . . . , xp−1 , make z1 the root and add the arcs (z1 , z2 ), (z2 , xp ). This
transforms T into an out-branching of D ′ with at most k vertices of reduced
degree.
Case 2. Now suppose that in T the vertices yi1 , . . . , yis ∈ Y have outneighbors in x1 , . . . , xp−1 . Since T is an out-branching, the set of outneighbors of yij and yil are disjoint for all j 6= l. In T , the out-neighbors
126
Chapter 7. The Directed Full Degree Spanning Tree Problem
of yij in {x1 , . . . , xp−1 } can be ordered in the natural way according to
their position in the path x1 → · · · → xp−1 . Let xqj be the first outneighbor of yij among {x1 , . . . , xp−1 } in T . Transform T into a digraph T1
by deleting out-arcs such that for 1 ≤ j ≤ s, the only out-neighbor of yij
among {x1 , . . . , xp−1 } is xqj . Sort the vertices yi1 , . . . , yis in increasing order
based on the order of the vertices xqj in the path x1 → · · · → xp−1 . Without
loss of generality we assume that the sorted order is also yi1 , . . . , yis .
Transform T1 yet again by connecting the vertices xi so that the resulting
digraph (which we continue to call T1 ) contains the paths:
• x0 → x1 → · · · → xq1 −1 ;
• yij → xqj → · · · → xqj+1 −1 , for 1 ≤ j ≤ s − 1;
• yis → xqs → · · · → xp−1 · · · .
Note that in T , the vertices xqj −1 , for 1 ≤ j ≤ s, are of out-degree zero.
Moreover the last path yis → xqs → · · · in the sequence contains all vertices xqs , . . . , xp−1 but may or may not contain the vertex xp . Observe
that T1 is an out-tree and that the only vertices y ∈ {yi1 , . . . , yis } whose
out-degree is reduced in this transformation had at least two out-neighbors
among the vertices {x1 , . . . , xp−1 } in T . Hence for every yij whose outdegree is reduced in transforming T to T1 , there exists a distinct vertex
in x1 , . . . , xp−1 of out-degree zero in T which is of full degree in T1 . Thus
the number of vertices of reduced degree does not change in this transformation.
Now delete the arcs (yij , xqj ), where 1 ≤ j ≤ s − 1, and add (xqj −1 , xqj ),
where 1 ≤ j ≤ s−1, so that the resulting digraph T2 contains the path x0 →
· · · → xqs−1 . In this transformation, the vertices which possibly have their
out-degree reduced are yij , for 1 ≤ j ≤ s − 1, but an equal number of
vertices xqj −1 , for 1 ≤ j ≤ s − 1, attain full degree. Therefore the number
of vertices of reduced out-degree in T2 is at most that in T1 . To obtain
an out-branching of D ′ from T2 , delete x1 , . . . , xqs−1 , add the arcs (yis , z1 ),
(z1 , z2 ) and connect z2 to the out-neighbor of xp−1 , if any. Note that this
transforms T2 into an out-branching of D ′ with at most k vertices of reduced
degree.
This completes the proof of the lemma.
It is easy to see that Rules 1 and 2 can be applied in O(n) time and
that Rule 3 can be applied in O(n + m) time. Note that Rule 3 is parameter
independent, that is, an application of the rule does not affect the parameter.
Consequently, it makes sense to talk about a digraph being reduced with
respect to Rule 3 as distinct from an instance of d-RDST being reduced with
respect to Rule 3. Our kernelization algorithm consists in applying Rules 1
to 3 repeatedly until the given instance is reduced.
7.4. d-RDST: A Problem Kernel
127
We next describe a lemma that we repeatedly make use of in the sequel.
Given a directed graph D, we let Vi (D) ⊆ V (D) denote the set of vertices
of out-degree i; V≥i (D) ⊆ V (D) denotes the set of vertices of out-degree at
least i.
Lemma 7.2. Let D be a directed graph reduced with respect to the Path
Rule (Rule 3) and let T be an out-branching of D with root r such that X
is the set of vertices of reduced out-degree. Then
|V (T )| ≤ 4|V0 (T ) ∪ V≥2 (T ) ∪ X| ≤ 4(|V0 (T )| + |X ∪ V0 (T )|).
Proof. If we view the out-branching T as an undirected graph, V0 (T ) is the
set of leaves and V≥2 (T ) is the set of vertices of degree at least three along
with the root r, if d+
T (r) ≥ 2. Thus V≥2 (T ) has at most one vertex of total
degree two and all other vertices are of total degree at least three. It is a
well-known fact that a tree with l leaves has at most l − 1 internal vertices
of degree at least three. Since V≥2 (T ) has at most one vertex of total degree
two, we have |V≥2 (T )| ≤ |V0 (T )|.
Now consider the vertices of the out-branching T which have out-degree
exactly one. Define W := X ∪ V0 (T ) ∪ V≥2 (T ) and let P be the set of
maximal dipaths in T such that for any dipath P = x0 → x1 → · · · → xp
in P we have that (1) d+
D (xi ) = 1 for 0 ≤ i ≤ p − 1, and (2) xp ∈ W .
Observe that every vertex with out-degree exactly one in T is contained
in exactly one path in P. Also observe that the set of vertices of outdegree exactly one in T not
P contained in W is precisely the set V1 (T ) \ X.
Therefore |V1 (T ) \ X| ≤ P ∈P (|P | − 1), where |P | denotes the number of
vertices in the path P . By Rule 3, any dipath P ∈ P has at most four
vertices and since the number of dipaths in P is at most |W |, we have
|V1 (T ) \ X| ≤ 3 · |P| ≤ 3 · |W | ≤ 3|X ∪ V0 (T ) ∪ V≥2 (T )|.
Since |V (T )| ≤ |V1 (T ) \ X| + |X ∪ V0 (T ) ∪ V≥2 (T )|, we have
|V (T )| ≤ 4|V0 (T ) ∪ V≥2 (T ) ∪ X| ≤ 4(|V0 (T )| + |V0 (T ) ∪ X|).
This completes the proof of the lemma.
We can now bound the size of a yes-instance of the d-RDST problem
on out-degree at least one digraphs that have been reduced with respect to
Rules 1 to 3.
Theorem 7.2. Let (D, k) be a yes-instance of the d-RDST problem on outdegree at least one digraphs that is reduced with respect to Rules 1 to 3.
Then |V (D)| ≤ 8k.
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Chapter 7. The Directed Full Degree Spanning Tree Problem
Proof. Since (D, k) is a yes-instance of the problem, let T be an outbranching of D and let X be the set of vertices of reduced degree in T ,
where |X| ≤ k. Every vertex of D is of out-degree at least one and
hence V0 ⊆ X, where V0 is the set of leaves in T . Consequently |X ∪ V0 | ≤ k
and |V0 | ≤ k and by Lemma 7.2, we have |V (T )| ≤ 8k, as claimed.
Observe that the crucial step in the proof above was to bound the number
of leaves in the solution out-branching. For out-degree at least one digraphs
this is easy since every leaf is a vertex of reduced degree. This is not the case
with general digraphs which may have an arbitrary number of vertices of
out-degree zero, all of which are of full degree in any out-branching. In the
next subsection we present a set of reduction rules for the d-RDST problem
in general digraphs which help us bound the number of vertices of out-degree
zero in terms of the parameter k.
7.4.2
An O(k 2 )-Vertex Kernel in General Digraphs
For general digraphs, we first consider an annotated version of the problem
as this seems to help in developing reduction rules. Eventually we will revert
to the original unannotated version. An instance of the annotated version
consists of a triplet (D, X, k) where D and k are, respectively, the input
digraph and the parameter, and X is a subset of V (D) such that in any outbranching with at most |X| + k vertices of reduced degree, the vertices of X
must be of reduced degree. The question in this case is to decide whether D
admits an out-branching where the set of vertices of reduced degree is X ∪S,
where S ⊆ V (D) \ X and |S| ≤ k. Call such an out-branching a solution
out-branching. To obtain a kernel for d-RDST, we apply the reduction rules
to an instance (D, k) after setting X = ∅.
Given an instance (D, X, k), we define the conflict set of a vertex u ∈
V (D) \ X as
C(u) := {v ∈ V (D) \ X : N + (u) ∩ N + (v) 6= ∅}.
Clearly vertices of out-degree zero have an empty conflict set. If a vertex v
has a non-empty conflict set then in any out-branching either v loses its
degree or every vertex in C(v) loses its degree. Moreover if u ∈ C(v) then v ∈
C(u) and in this case we sayP
that u and v are in conflict. The conflict number
of D is defined as c(D) := v∈V (D)\X |C(v)|.
We assume that the input instance is (D, X, k) and the kernelization
algorithm consists in applying each reduction rule repeatedly, in the order
given below, until no longer possible. Therefore when we say that Rule i is
indeed a reduction rule we assume that the input instance is reduced with
respect to the rules preceding it.
Rule 0. If u ∈ X and d+ (u) = 1, delete the out-arc from u and return (D, X, k).
7.4. d-RDST: A Problem Kernel
129
The vertices in X are of reduced degree in any solution out-branching.
Thus if a vertex in X has out-degree exactly one, this out-arc will never be
part of a solution out-branching, and deleting it will not change the solution
structure.
Rule 1. If there exists u ∈ V (D) such that the number of in-neighbors of
u in V (D) \ X is at least k + 2 then return no; else return (D, X, k).
In the last subsection, we already showed that this rule is indeed a reduction rule.
Rule 2. If u ∈ V (D) \ X and |C(u)| > k, set X ← X ∪ {u} and k ←
k − 1. Furthermore if d+ (u) = 1 then delete the out-arc from u and
return (D, X, k).
If the conflict set C(u) of u ∈ V (D) \ X is of size at least k + 1 and if u
is of full degree in some out-branching T , then every vertex in C(u) must be
of reduced degree in T . Therefore if (D, X, k) is a yes-instance then u must
have its degree reduced in every solution out-branching. In addition, if u
has out-degree exactly one, then this out-arc cannot be part of any solution
out-branching and can be deleted. This shows that Rule 2 is a reduction
rule.
Rule 3. If c(D) > 2k2 then return no, else return (D, X, k).
Lemma 7.3. Rule 3 is a reduction rule for the d-RDST problem.
Proof. To see why Rule 3 qualifies to be a reduction rule, construct the
conflict graph CD,X of the instance (D, X, k) which is defined as follows.
The vertex set V (CD,X ) := V (D) \ X and two vertices in V (CD,X ) have
an edge between them if and only if they are in conflict. Since the size of
the conflict set of any vertex is at most k (as D is reduced with respect to
Rule 2), the degree of any vertex in CD,X is at most k. The key observation
is that if T is any solution out-branching of (D, X, k) in which the set of
vertices of reduced degree is X ∪ S with S ⊆ V (D) \ X, then S forms
a vertex cover of CD,X . Since we require that |S| ≤ k, the number of
edges in CD,X is at most k2 . For a vertex v ∈ V (D) \ X, let d′ (v) be the
number of
v in the conflict graph CD,X . Observe that
Pneighbors of vertexP
c(D) := v∈V (D)\X |C(v)| = v∈V (D)\X d′ (v) ≤ 2k2 . The last inequality
follows from the fact that sum of degrees of vertices in a graph is equal to
twice the number of edges.
Rule 4. If u ∈ V (D) such that d+ (u) = 0 and d− (u) = 1 then delete u
from D and return (D, X, k).
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Chapter 7. The Directed Full Degree Spanning Tree Problem
It is easy to see that Rule 4 is a reduction rule: vertex u is of full degree
in any solution and it does not determine whether its parent is of full or
reduced degree in a solution out-branching and therefore can be deleted. To
obtain a solution out-branching for D from a solution T ′ for D \ v, simply
add the arc between u and its parent in T ′ .
Rule 5. Let u ∈ V (D) be of out-degree zero and let v1 , . . . , vr be its inneighbors, where r > 2. Delete u and add 2r new vertices uij ,
where 1 ≤ i < j ≤ r; for a newly added vertex uij add the arcs
(vi , uij ) and (vj , uij ). Return (D, X, k).
Note that vertex u forces at least r − 1 vertices from {v1 , . . . , vr } to be
of reduced degree in any out-branching
of D. This situation is captured by
deleting u and introducing 2r vertices as described in the rule. These 2r
vertices then force at least r − 1 vertices from {v1 , . . . , vr } to be of reduced
degree in any out-branching of the transformed graph. The upshot is that
each vertex of out-degree zero has in-degree exactly two.
Lemma 7.4. Rule 5 is a reduction rule for the d-RDST problem.
Proof. Let (D, X, k) and (D ′ , X, k) be the instances of d-RDST before and
after one application of Rule 5, respectively. We claim that (D, X, k) is a
yes-instance if and only if (D ′ , X, k) is a yes-instance.
Let T be an out-branching of D that certifies that (D, X, k) is a yesinstance. Then at least r − 1 vertices from {v1 , . . . , vr } are of reduced
degree in T . Transform T into an out-branching T ′ for (D ′ , X, k) as follows. Delete u from T and introduce the vertices uij for 1 ≤ i < j ≤ r.
If vi ∈ {v1 , . . . , vr } was of full degree in T then in T ′ add the arcs (vi , upq )
for all 1 ≤ p < q ≤ r; otherwise add the arcs (v1 , upq ) for all 1 ≤ p < q ≤ r.
The out-branching T ′ certifies that (D ′ , X, k) is a yes-instance.
Conversely suppose that the out-branching T ′ certifies that (D ′ , X, k)
is a yes-instance. Again at least r − 1 vertices from {v1 , . . . , vr } are of
reduced degree in T ′ . Transform T ′ into an out-branching T for (D, X, k)
as follows. Delete the vertices uij for 1 ≤ i < j ≤ r and introduce vertex u.
If vi ∈ {v1 , . . . , vr } was of full degree in T ′ , add the arc (vi , u) in T ; otherwise
add the arc (v1 , u). Clearly T certifies that (D, X, k) is a yes-instance.
Rule 6. If u, v ∈ V (D) \ X have p > 1 common out-neighbors of out-degree
zero, delete all but one of them. Return (D, X, k).
Rule 7. If u ∈ V (D) is of out-degree zero such that at least one in-neighbor
of u is in X, delete u. Return (D, X, k).
By Rule 5, it is clear that if u, v ∈ V (D) \ X have at least two common
out-neighbors of out-degree zero then these out-neighbors have in-degree
exactly two. It is intuitively clear that these out-neighbors are equivalent in
7.4. d-RDST: A Problem Kernel
131
some sense and it suffices to preserve just one of them. It is easy to show
that the original instance has a solution out-branching if and only if the
instance obtained by one application of Rule 6 has a solution out-branching.
As for Rule 7, if u has two in-neighbors v and w and if v ∈ X, we can delete
the arc (v, u) without altering the solution structure. But then v is a private
neighbor of w of out-degree zero and hence can be deleted by Rule 4.
Rule 8 (The Path Rule). Let x0 , x1 . . . , xp−1 , xp be a sequence of vertices in D such that for 0 ≤ i ≤ p − 1 we have d+ (xi ) = 1 and
(xi , xi+1 ) ∈ A(D). Let Y0 be the set of in-neighbours of x1 , . . . , xp−1
and let Y := Y0 \ {x0 , x1 , . . . , xp−2 }. Delete the vertices x1 , . . . , xp−1
and add two new vertices z1 , z2 and the arcs (x0 , z1 ), (z1 , z2 ), (z2 , xp ).
For y ∈ Y , if y has at least two out-neighbors in {x1 , . . . , xp−1 },
then add arcs (y, z1 ), (y, z2 ); if y has exactly one out-neighbor in
{x1 , . . . , xp−1 }, then add the arc (y, z1 ). Return (D, k).
This is Rule 3 from the previous subsection where it was shown to be a
reduction rule for the d-RDST problem (note that the proof of Lemma 7.1
did not use the fact that the input was an out-degree at least one digraph).
By Rule 0, no vertex on the path x0 , x1 , . . . , xp−1 is in X and therefore the
proof of Lemma 7.1 continues to hold for the annotated case as well.
It is easy to see that a single application of Rules 5 or 6 takes time O(n2 );
all other rules take time O(n + m). We are now ready to bound the number
of vertices of out-degree zero in a reduced instance of the annotated problem.
Lemma 7.5. Let (D, X, k) be a yes-instance of the annotated d-RDST problem that is reduced with respect to Rules 0 through 8 mentioned above. Then
the number of vertices of out-degree zero in D is at most k 2 .
Proof. Let u be a vertex of out-degree zero. By Rules 4 and 5, it must have
exactly two in-neighbors, say, x and y. By Rule 7, neither x nor y is in X
and are therefore still in conflict in the reduced graph. Hence, either x or y
must be of reduced degree in any solution out-branching. Furthermore any
vertex not in X can have at most k out-neighbors of out-degree zero since,
by Rule 2, any vertex not in X is in conflict with at most k other vertices
and, by Rule 6, two vertices in conflict can have at most one common outneighbor of out-degree zero. Since (D, X, k) is assumed to be a yes-instance,
at most k vertices can lose their out-degree in any solution out-branching.
Moreover, by Rule 4, any vertex of out-degree zero is an out-neighbor of at
least one vertex of reduced degree. Therefore the total number of vertices
of out-degree zero is at most k2 .
Lemma 7.6. Let (D, k) be a yes-instance of the d-RDST problem and suppose that (D1 , X, k1 ) is an instance of the annotated d-RDST problem reduced with respect to Rules 0 through 8 by repeatedly applying them on (D, k),
by initially setting X = ∅. Then |V (D1 )| ≤ 8(k2 + k).
132
Chapter 7. The Directed Full Degree Spanning Tree Problem
Proof. Since reduction rules map yes-instances to yes-instances and does
not allow the parameter to increase, it is clear that (D1 , X, k1 ) is a yesinstance of the annotated d-RDST problem and that k1 +|X| ≤ k. Therefore
let T1 be a solution out-branching of (D1 , X, k1 ). A leaf of T1 is either a
vertex of out-degree zero in D1 or a vertex of reduced degree. By Lemma 7.5,
the total number of vertices of out-degree zero in D1 is at most k12 ≤ k2 and
since T1 is a solution out-branching, the total number of vertices of reduced
degree is at most k1 + |X| ≤ k. Thus the number of leaves of T1 is at
most k2 + k and by Lemma 7.2 we have |V (T1 )| ≤ 4(k2 + k + k2 + k) =
8(k2 + k).
We now show how to obtain a kernel for the original (unannotated)
version of the problem. Let (D, k) be an instance of the d-RDST problem
and let (D ′ , X, k′ ) be the instance obtained by applying reduction rules 0
through 8 on (D, k) until no longer possible, by initially setting X = ∅.
By Lemma 7.6, we know that if (D, k) is a yes-instance then |V (D ′ )| ≤
8(k2 + k) and that k′ + |X| = k. To get back an instance of the unannotated
version, apply the following transformation on (D ′ , X, k′ ). If X 6= ∅, add a
directed path Y = y1 , . . . , yk+2 to D ′ and for x ∈ X add the out-arc (x, yi )
for 1 ≤ i ≤ k + 2. Call the resulting digraph D ′′ .
We claim that (D ′ , X, k′ ) has a solution out-branching T ′ with at most
|X| + k′ vertices of reduced degree and where all vertices in X have their
degree reduced if and only if D ′′ admits an out-branching with at most k
vertices of reduced degree. Let T ′ be a solution out-branching for (D ′ , X, k ′ ).
To obtain a solution out-branching T ′′ of D ′′ simply add the path Y to T ′
and the out-arc (x1 , y1 ). Clearly T ′′ has at most k′ + |X| vertices of reduced
degree. Conversely let T ′′ be a solution out-branching for D ′′ . First note
that every vertex in X must be of reduced degree in T ′′ . For if x ∈ X is of full
degree then k + 1 vertices y1 , . . . , yk+1 are of reduced degree, contradicting
the fact that T ′′ has at most k vertices of reduced degree. Therefore we may
assume that, in T ′′ , vertex x1 has an out-arc to the start vertex y1 of the
path y1 , . . . , yk+2 which appears as is in the out-branching. That is, we may
assume that the vertices in the path Y are always of full degree in any outbranching and that there exists at most k vertices in V (D ′′ ) \ (V (Y ) ∪ X) of
reduced degree in T ′′ . To obtain a solution out-branching T ′ for (D ′ , X, k ′ )
simply delete the path Y from T ′′ .
Since we add at most k + 2 vertices in this transformation, we have
Theorem 7.3. The d-RDST problem, parameterized by the number of vertices of reduced degree, admits a problem kernel with at most 8k2 + 9k + 2
vertices.
7.5. An Algorithm for the d-RDST Problem
7.5
133
An Algorithm for the d-RDST Problem
In this section we describe a branching algorithm for the d-RDST problem
with running time O(5.942k · nO(1) ). We first observe that in order to construct a solution out-branching of a given digraph, it is sufficient to know
which vertices will be of reduced degree.
Lemma 7.7. Let D = (V, A) be a digraph and let X be the set of vertices
of reduced degree in some out-branching of D. Given D and X, one can
in polynomial time construct an out-branching of D in which the vertices of
reduced degree is a subset of X.
Proof. We describe an algorithm that constructs such an out-branching of D.
Given D and X, our algorithm first constructs a digraph D ′ with vertex
set V (D) in which
1. all vertices in V (D) \ X are connected to their out-neighbors in D by
solid arcs;
2. a vertex x ∈ X has a dotted out-arc to a vertex y if (x, y) ∈ A(D)
and y has no solid in-arc in D ′ .
We are guaranteed that there exists an out-branching of D ′ in which all
solid arcs are present but in which one or more dotted arcs may be missing.
Note that in D ′ , a vertex with a solid in-arc has no other (solid or dotted)
in-arcs.
Our algorithm now runs through all possible choices of the root of the
proposed out-branching. For each choice of root, it does a modified breadthfirst search (BFS) starting at the root. In the modified BFS-routine, when
the algorithm visits a vertex v, it solidifies all dotted out-arcs from v, if any.
For each dotted arc (v, w) that it solidifies, it deletes all dotted in-arcs to w.
The algorithm then inserts the out-neighbors of v in the BFS-queue. If the
BFS-tree thus constructed includes all vertices of D ′ , the algorithm outputs
this out-branching, or else, moves on to the next choice of root.
Claim. Suppose that r is the root of an out-branching of D ′ in which X is
the vertex-set of reduced degree. Then the above algorithm, on selecting r
as root, succeeds in constructing an out-branching in which the vertices of
reduced degree is a subset of X.
In order to prove this claim, it is sufficient to show that in the BFS-tree T
constructed by the algorithm with r as root, every vertex of D ′ is reachable
from r. This suffices because every vertex in V (D ′ ) \ X is of full degree in T .
Therefore let v be a vertex not reachable from r such that the distance,
in D ′ , from v to r is the shortest among all vertices not reachable from r
in T . Let r, v1 , . . . , vl , v be a shortest dipath from r to v in D ′ . By our
choice of v, all vertices v1 , . . . , vl are reachable from r in T . Note that the
134
Chapter 7. The Directed Full Degree Spanning Tree Problem
arc (vl , v) must have been dotted and in fact all in-arcs to v were dotted
in D ′ . When the algorithm visited vl , the only reason it could not solidify
the arc (vl , v) must have been because v already had a solid in-arc into it
and hence the arc (vl , v) had already been deleted. Suppose that v has a
solid in-arc from u. Then u must have already been visited before vl at
which time the dotted arc (u, v) was solidified. But this means that u, and
hence v, is reachable from r in the BFS-tree T , a contradiction.
We now have an O(kO(k) · nO(1) ) algorithm for the d-RDST problem:
Given an instance (D, k), we first obtain a kernel of size O(k2 ) using Theorem 7.3 and then run over all possible vertex-subsets X of the kernel of size
at most k to determine the set of vertices of reduced degree. Then using
Lemma 7.7, we verify whether one can indeed construct an out-branching
in which the set of vertices of reduced degree is X.
RDST (D, X, k)
Input: A digraph D = (V, A); X ⊆ V , such that the vertices in X will be
of reduced degree in the out-branching that is being constructed; an integer
parameter k. The algorithm is initially called after setting X = ∅.
Output: An out-branching of D in which every vertex of X is of reduced degree
and with at most k vertices of reduced degree in total, if one exists, or no,
signifying that no such out-branching exists.
1. If k < 0 or |X| > k return no.
2. If no two vertices in V (D) \ X have a common out-neighbor then
(a) Reduce (D, X, k) with respect to Rules 1′ through 5′ .
(b) For each (k − |X|)-sized subset Y of V (D) \ X, check if there exists
an out-branching of D in which the vertex set of reduced degree
is X ∪ Y . If yes, then “expand” this out-branching to an outbranching for the original instance and return the solution; else
return no.
3. Let u, v ∈ V (D) \ X be two vertices with a common out-neighbor then
(a) X ← X ∪ {u}; Z = Call RDST(D, X, k − 1).
(b) If Z 6= no then return Z.
(c) X ← X ∪ {v}; Return RDST(D, X, k − 1).
Figure 7.3: Algorithm RDST.
In the rest of this section, we give an improved algorithm with running
time O(ck · nO(1) ), for a constant c. Our algorithm (see Figure 7.3) is based
on the simple observation that if two vertices u and v of the input digraph D
have a common out-neighbor then one of them must be of reduced degree in
any out-branching of D. The algorithm recurses on vertex-pairs that have a
7.5. An Algorithm for the d-RDST Problem
135
common out-neighbor and, along each branch of the recursion tree, builds a
set X of vertices which would be the candidate vertices of reduced degree in
the out-branching that it attempts to construct. When there are no vertices
to branch on, it reduces the instance (D, X, k) with respect to the following
rules.
Rule 1′ . If u ∈ X and d+ (u) = 1, delete the out-arc from u and return (D, X, k). This is Rule 0 from Section 7.4.2.
Rule 2′ . Let u ∈ V (D) be of out-degree zero and let v1 , . . . , vr be its inneighbors. If vi ∈ X for all 1 ≤ i ≤ r, assign v1 as the parent of u and
delete u. If there exists 1 ≤ i ≤ r such that vi ∈
/ X then assign vi as
the parent of u and delete u. Return (D, X, k).
Rule 3′ . This is Rule 8 from Section 7.4.2.
Rule 1′ is a reduction rule because a vertex of out-degree exactly one
that is of reduced degree must necessarily lose its only out-arc. As for
Rule 2′ , we know that in the instance (D, X, k) obtained after the algorithm
finishes branching, no two vertices of V (D)\X have a common out-neighbor
and therefore at least r − 1 in-neighbors of u must be in X (and of reduced
degree). If all in-neighbors of u are of reduced degree, we arbitrarily fix one of
them as parent of u (so that we can construct an out-branching of the original
instance later on) and delete u. If exactly r − 1 in-neighbors of u are already
of reduced degree, we choose that in-neighbor not in X as the parent of u and
delete u. Also note that when applying Rule 3′ to a path x0 , x1 , . . . , xp−1 , xp ,
the vertices x0 , x1 , . . . , xp−1 are not in X, by Rule 1′ . Therefore if Y is the set
of in-neighbors of x1 , . . . , xp−1 , excluding {x0 , x1 , . . . , xp−2 }, then Y ⊆ X.
Observe the following:
1. By Rule 2′ , no vertex in the reduced instance (D, X, k) has out-degree
zero.
2. Every vertex in the subdigraph induced by V (D) \ X has in-degree at
most one and hence each connectivity component (a connected component in the undirected sense) is either a dicycle, or an out-tree or
a dicycle which has out-trees rooted at its vertices. Thus each connectivity component has at most one dicycle and if a component does
have a dicycle then it can be transformed into an out-branching by
deleting an arc from the cycle. Such a digraph is called a pseudo
out-forest [118].
We now reduce the instance (D, X, k) with respect to the following two
rules:
Rule 4′ . If at least k + 1− |X| connectivity components of D[V \X] contain
dicycles, then return no; else return (D, X, k).
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Chapter 7. The Directed Full Degree Spanning Tree Problem
Rule 5′ . If a connectivity component of D[V \ X] is a dicycle C such that
no vertex in V (C) has an out-neighbor in X, pick a vertex u ∈ X
with an arc to C and fix it as the “entry point” to C; delete C and
set k ← k − 1; return (D, X, k).
Rule 4′ is a reduction rule as every connectivity component that has a
dicycle contains at least one vertex that will be of reduced degree. If the
number of such components is at least k + 1 − |X|, one cannot construct an
out-branching with at most k vertices of reduced degree where all vertices
in X have their degree reduced. To see that Rule 5′ is a reduction rule,
first note that since C has no out-arcs, it cannot contain the root of the
proposed out-branching. Any path from the root to C must necessarily
include a vertex from X and it does not matter which arc out of X we
use to get to C, since every vertex in X has its degree reduced anyway.
Moreover, in any out-branching, exactly one vertex of C must be of reduced
degree. Therefore if (D ′ , X ′ , k′ ) is the instance obtained by one application
of Rule 5′ to the instance (D, X, k), then it is easy to see that these instances
must be equivalent. Also note that each application of Rule 1′ through 5′
takes time O(n + m).
Lemma 7.8. Let (D, X, k) be an instance of the d-RDST problem in which
no two vertices of V (D) \ X have a common out-neighbor, and reduced with
respect to Rules 1′ through 5′ . Then |V (D) \ X| ≤ 7|X|.
Proof. Let D ′ be a digraph obtained from D by deleting all out-arcs from
the vertices in X. Therefore in D ′ , every vertex of X has out-degree zero
and in-degree at most one. We show that a connectivity component of D ′
that has p vertices of X has at most 7p vertices of V (D ′ ) \ X. This will
prove the lemma.
If a connectivity component of D ′ is an out-tree T ′ , then every leaf of
this out-tree is a vertex of X. If T ′ has p leaves, then applying Lemma 7.2
to T ′ , we have that |V (T ′ )| ≤ 8p. Since exactly p of these vertices are
from X, the number of vertices of V (D ′ ) \ X in the out-tree is at most 7p.
Therefore let R be a connectivity component of D ′ containing a dicycle such
that |V (R) ∩ X| = p. Then R has exactly one dicycle, say C. By Rule 5′ , C
has a vertex x with an out-neighbor in V (R)\V (C), and therefore d+
R (x) ≥ 2.
Let y be the out-neighbor of x in C. Delete the arc (x, y) to obtain an outbranching T with root y. Note that the number of leaves in T is the same
as that in R. Moreover in transforming R to T , only one vertex (namely x)
loses its out-degree. By Lemma 7.2,
|V (T )| ≤ 4|x ∪ V0 (T ) ∪ V≥2 (T )| ≤ 4|V0 (T )| + 4|x ∪ V≥2 (T )|,
and since |x ∪ V≥2 (T )| ≤ 1 + (|V0 (T )| − 1) = p, we have |V (T )| ≤ 8p.
Consequently |V (R) \ X| ≤ 7p. This completes the proof of the lemma.
7.5. An Algorithm for the d-RDST Problem
137
To construct an out-branching, it is sufficient to choose the remaining k−
|X| vertices of reduced degree from the vertices in V (D)\X. Setting |X| = c,
the exponential term in the running time of the algorithm is bounded above
by the function
k
X
7c
7c
c
c
2 ·
≤ k · max 2 ·
.
0≤c≤k
k−c
k−c
c=0
7c We will show that the function h(k) := max0≤c≤k 2c · k−c
is bounded above
by (k + 1) · 5.942k .
Theorem 7.4. Given a digraph D and a nonnegative integer k, one can
decide whether D has an out-branching with at most k vertices of reduced
degree, and if so, construct such an out-branching in time O(5.942k · k2 · (n +
m)).
Finally we prove the claimed upper-bound for the function h(k). We
first need a lemma.
Lemma 7.9. For nonnegative integers k ≤ n and any real x > 0,
n
(1 + x)n
.
≤
xk
k
Proof. Since
x
where ∆ =
P
−k
k
(1 + x) = x
n i−k
0≤i6=k≤n i x
−k
·
n X
n
i=0
n
x =
+ ∆,
i
k
i
≥ 0, the result follows immediately.
Lemma 7.10. For a nonnegative integer k, the function
7c
h(k) = max 2c ·
0≤c≤k
k−c
is bounded above by (k + 1) · 5.942k .
Proof. Using the bound in Lemma 7.9, we have that for any 0 ≤ c ≤ k and
any x > 0,
c
7c
2x(1 + x)7
7c
c
c (1 + x)
2 ·
≤2 ·
=
.
k−c
xk−c
xk
Since the above inequality holds for any x > 0, it holds, in particular, for the
positive roots of the equation 2x(1 + x)7 − 1 = 0. This equation has exactly
one positive root and its value is approximately 0.16830. Substituting this
value in the previous inequality, we obtain
7c
1
c
≤ 5.942k ,
2 ·
≤
0.16830k
k−c
and since there are k + 1 such terms in h(k), the value of h(k) is at most (k +
1) · 5.942k .
138
7.6
Chapter 7. The Directed Full Degree Spanning Tree Problem
Concluding Remarks
We studied a natural generalization of the Full Degree Spanning Tree
problem to directed graphs. We showed that the d-FDST problem is W[1]hard even on the class of DAGs and that the d-RDST problem is fixedparameter tractable. For the d-RDST problem, we obtained a kernel with
at most 8k2 + 9k + 2 vertices and an algorithm with running time O(5.942k ·
(n+m)). Natural open questions are to investigate whether d-RDST admits
a linear-vertex kernel and design algorithms with better running times.
Chapter 8
Summary and Future Research
In this thesis, we studied different parameterizations of NP-optimization
problems with the objective of identifying parameterizations that are most
likely to be useful in practice.
We started out in Chapter 2 with NP-optimization problems that are
fixed-parameter tractable to begin with. The parameter in such cases always assumes a value that is a significant fraction of the input size and we
argued that such a parameterization is not useful in practice since any FPTalgorithm for the problem runs in time that is essentially exponential in the
input-size. For such problems the natural parameter is the difference between the optimum and the guaranteed lower or upper-bound. In Chapter 3
we tried to extend these ideas to the realm of approximation algorithms.
From Chapters 4 through 7, we saw several situations where an NPoptimization problem is parameterizable in more than one way. If the problem at hand is W-hard for one parameter, we tried a different parameterization for which it is in FPT. Such a strategy gives us an idea as to the
best way to tackle the intractability of a problem. In this final chapter we
summarize the results presented in this thesis and collate the open problems
that appear throughout this thesis.
8.1
Parameterizing Problems Beyond Default Values
We showed that there exist many NP-optimization problems with the property that their optimum solution size is bounded below by an unbounded
function of the input size. The standard parameterized version of these
problems is not interesting as they are trivially in FPT. We argued that the
natural way to parameterize such problems is to let the parameter represent
the deficit between the optimum and the lower bound, that is, to parameterize above the guaranteed lower bound. But we also observed that the
lower bound must be tight in some sense in order for the above-guarantee
parameterizations to be interesting. This led us to introduce tight lower and
139
140
Chapter 8. Summary and Future Research
upper bounds. We showed that parameterizations of the type tlb+ǫ·|I|+k
or tub − ǫ ·|I|− k, that are sufficiently beyond tight bounds, are not in FPT,
unless P = NP.
There are quite a few open problems. Firstly we would like to have
general results about classes of NP-optimization problems whose optima
are bounded below (above) by a nontrivial lower (upper) bound.
Open Problem 8.1. Is there a characterization for the class of problems
for which the above or below-guarantee question with respect to a tight
lower or upper bound is in FPT (or W[1]-hard)?
There are several directions to pursue from an algorithmic point of view.
Not many results are known on parameterized above or below-guarantee
problems and, in particular, the complexity of problems (1) through (7) in
Chapter 2 Section 2.4, when parameterized above their guaranteed values,
is open. Some of the more interesting above-guarantee problems are:
Open Problem 8.2. Planar Independent Set: Given an n-vertex planar graph and an integer parameter k, does G have an independent set of
size at least ⌈n/4⌉ + k?
Open Problem 8.3. Max Exact c-Sat: Given a Boolean CNF formula F
with m clauses such that each clause has exactly c distinct literals and
an integer parameter k, does there exist an assignment that satisfies at
least (1 − 2−c )m + k clauses
Recently Gutin et al. [74] have shown that Max Lin-2, Max Acyclic
Subgraph and a special case of Max Exact c-Sat are in FPT. In an
another paper [75], the same authors show that Max 2-Sat is in FPT when
parameterized above the bound 3m/4 by demonstrating a kernel with O(k 2 )
variables. Finally in 2010, Alon et al. [4] showed that Max Exact c-Sat
is in FPT and that it admits a kernel with O(k 2 ) variables.
Here is an interesting below-guarantee problem whose parameterized
complexity is open. Recall that an undirected graph G is perfect if for
every induced subgraph H of G, the chromatic number and clique number
of H are equal.
Open Problem 8.4. [36, 105] Perfect Vertex Deletion: Let G be a
graph on n vertices and m edges and k a nonnegative integer. Does there
exist a vertex-induced subgraph on n − k vertices that is perfect? A similar
question can be framed for the edge version. What is the approximability
of this problem?
8.2
Kőnig Subgraph Problems
We studied a set of subgraph problems, called Kőnig Subgraph problems,
which are defined as follows:
8.3. Unique Coverage
141
1. Kőnig Vertex (Edge) Deletion (KVD/KED). Given a graph G
and a nonnegative integer k, does there exist at most k vertices (respectively, edges) whose deletion results in a Kőnig subgraph?
2. Vertex (Edge) Induced Kőnig Subgraph (VKS/EKS). Given
a graph G and a nonnegative integer k, does there exist at least k
vertices (respectively, edges) which induce a Kőnig subgraph?
We showed that KVD is in FPT whereas VKS is W[1]-hard. The EKS
problem is in FPT and we conjecture that KED is W[1]-hard.
Open Problem 8.5. What is the parameterized complexity of the Kőnig
Edge Deletion problem?
Our algorithm for KVD uses the FPT-algorithm for Above Guarantee
Vertex Cover as a subroutine, which in turn, uses the one for Min 2-Cnf
Del developed by Razgon and O’Sullivan [116].
Open Problem 8.6. Design FPT-algorithms for Kőnig Vertex Deletion and Above Guarantee Vertex Cover with a run-time better
than O(15k · k2 · |E|3 ), perhaps without using the algorithm for Min 2-Cnf
Del.
With regards to polynomial-time approximability, we show that if the
Unique Games Conjecture is true, the problems KVD, KED, and Min
2-Cnf Del do not admit constant-factor approximation algorithms; VKS is
as hard to approximate as Independent Set; and EKS admits a factor-5/3
algorithm.
We also showed that any graph G = (V, E) with maximum matching M
has an edge-induced Kőnig subgraph of size at least
where EM
3|EM | |E − EM | |M |
+
+
,
4
2
4
is the set of edges in the graph G[V (M )].
Open Problem 8.7. Is this lower bound tight? What is the parameterized
complexity of the following above-guarantee Edge Induced Kőnig Subgraph problem: Given a graph G = (V, E) with maximum matching M
and an integer parameter k, does G have an edge-induced Kőnig subgraph
of size at least
3|EM | |E − EM | |M |
+
+
+ k?
4
2
4
8.3
Unique Coverage
We considered several plausible parameterizations of the Unique Coverage problem and showed that except for the standard parameterized version (and a generalization of it), all other versions are unlikely to be fixedparameter intractable. We also described problem kernels for several special
142
Chapter 8. Summary and Future Research
cases of the standard parameterized version of Unique Coverage and
showed that the general version admits a kernel with at most 4k sets. We
noted that this is essentially the best possible kernel due to a lower bound
result by Dom et al. [46].
We gave FPT-algorithms for Budgeted Unique Coverage and several of its variants using the well-known method of color-coding. Our randomized algorithms have good running times but the deterministic algorithms make use of perfect hash families and this introduces large constants
in the running times, a common enough phenomenon when derandomizing
randomized algorithms using such function families [6]. It would be interesting to obtain faster algorithms.
Open Problem 8.8. Is there an FPT-algorithm for Unique Coverage
with run-time O∗ (2O(k) )? For Budgeted Unique Coverage, is there an
algorithm with run-time O ∗ (2O(k log log B) )?
For derandomizing our algorithms, we used (k, s)-hash families. We also
showed the existence of such families of size (2e)sk/t ·s log n, which is smaller
than that of the best-known (n, t, s)-perfect hash families.
Open Problem 8.9. Give an explicit construction of an (k, s)-hash family
with domain [n], range [t] and of size O(2O(sk/t) · s log n).
We also showed that given an instance (U , F, k) of the Unique Coverage problem, one can always cover at least |U |/8 log M elements, where M =
maxS∈F |S|.
Open Problem 8.10. Is this lower bound tight? What is the parameterized
complexity of the above-guarantee Unique Coverage problem: given an
instance (U , F, k), does there exist a subfamily F ′ of F that uniquely covers
at least |U |/8 log M + k elements, where M = maxS∈F |S|?
8.4
NP-Optimization Problems on Directed Graphs
We considered two problems on directed graphs. The first problem was
that of finding an induced subdigraph with k vertices in a given digraph
that satisfied some fixed hereditary property. We completely characterized
hereditary properties for which this problem is in FPT or W[1]-complete.
We first did this for general directed graphs and then for oriented graphs.
We also characterized hereditary properties for which the induced subgraph
problem is W[1]-complete in general directed graphs but in FPT for oriented
graphs.
In this context, we consider a related problem called the Graph Modification problem P(i, j, k) which asks whether a given input graph G
can be ‘modified’ by deleting at most i vertices, j edges and adding at
8.5. Concluding Remarks
143
most k edges so that the resulting graph satisfies property P. Cai [25]
has shown that if a hereditary property P has a finite forbidden set, the
graph modification problem P(i, j, k) is fixed-parameter tractable with parameters i, j, k. There is no general result for the case when the forbidden
set is infinite. For instance, the Odd Cycle Transversal problem is
fixed-parameter tractable [117] whereas the Wheel-Free Vertex (Edge)
Deletion problem is W[2]-hard [95].
Open Problem 8.11. Is there a characterization of hereditary properties
with an infinite forbidden set for which the Graph Modification problem
is either in FPT (or W-hard)?
The other problem on directed graphs that we studied is a generalization
of the (undirected) Full-Degree Spanning Tree problem. We showed
that the d-FDST problem is W[1]-hard even on the class of DAGs and that
the d-RDST problem is fixed-parameter tractable. For the d-RDST problem,
we obtained a kernel with at most 8k2 + 9k + 2 vertices and an algorithm
with run-time O(5.942k · nO(1) ).
Open Problem 8.12. Is there a sub-quadratic kernel for d-RDST? Design
an FPT-algorithm for d-RDST with a run-time better than O(5.942k ·nO(1) ).
8.5
Concluding Remarks
Parameterized Complexity has emerged as an important discipline both from
a theoretical and a practical perspective. The study of fixed-parameter algorithms and kernelization, in particular, has led to new insights into the
hardness of NP-complete problems and has fostered the development of interesting algorithmic techniques that have proved useful in practice. In this
thesis we tried to address issues relating to feasible parameterizations of
problems that may be useful in practice. As we have seen in this chapter, there are many open questions. With the rapid development that this
subject is witnessing, we can expect at least some of them to be resolved
satisfactorily in the near future.
144
Chapter 8. Summary and Future Research
Publications
1. M. Mahajan, V. Raman and S. Sikdar. Parameterizing NP-optimization
Problems Above or Below Guaranteed Values. Journal of Computer and
System Sciences, Volume 75, Issue 2, Pages 137-153, 2009. A preliminary
version appeared in Proceedings of the 2nd International Workshop on Parameterized and Exact Computation (IWPEC 2006), Springer-Verlag, LNCS
Volume 4169, Pages 38-49, 2006.
2. V. Raman and S. Sikdar. Approximating Beyond the Limit of Approximation.
Manuscript.
3. S. Mishra, V. Raman, S. Saurabh, S. Sikdar and C.R. Subramanian. The
Complexity of Kőnig Subgraph Problems and Above-Guarantee Vertex Cover.
To appear in Algorithmica. This paper was an amalgamation of the following
two papers.
(a) S. Mishra, V. Raman, S. Saurabh, S. Sikdar and C. R. Subramanian.
The Complexity of Finding Subgraphs whose Matching Number Equals
the Vertex Cover Number. In Proceedings of the 18th International
Symposium on Algorithms and Computation (ISAAC 2007), Springer
LNCS Volume 4835, Pages 268-279, 2007.
(b) S. Mishra, V. Raman, S. Saurabh and S. Sikdar. Kőnig Deletion
Sets and Vertex Covers Above the Matching Size. In Proceedings of
the 19th International Symposium on Algorithms and Computation
(ISAAC 2008), Springer LNCS Volume 5369, Pages 836-847, 2008.
4. H. Moser, V. Raman and S. Sikdar. The Complexity of the Unique Coverage
Problem. In Proceedings of the 18th International Symposium on Algorithms
and Computation (ISAAC 2007), Springer LNCS Volume 4835, Pages 621631, 2007.
5. N. Misra, V. Raman, S. Saurabh and S. Sikdar. Budgeted Unique Coverage
and Color-Coding. In Proceedings of the 4th Computer Science Symposium
in Russia, CSR 2009, Springer LNCS, Volume 5675, Pages 310-321, 2009.
6. V. Raman and S. Sikdar. Parameterized Complexity of the Induced Subgraph
Problem in Directed Graphs. Information Processing Letters, Volume 104,
Pages 79-85, 2007.
7. D. Lokshtanov, V. Raman, S. Saurabh and S. Sikdar. On the Directed Degree
Preserving Spanning Tree Problem. In Proceedings of the Fourth International Workshop on Parameterized and Exact Computation (IWPEC 2009),
Springer LNCS, Volume 5917, Pages 276-287, 2009.
145
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